Hello. This is my first guide ever.
Some of you might know me as ianto525.
I am completely open for arguments, but I'll place my reasoning inside each choice if there is an alternate in detail. If I do find a just-as-good or better alternative, I'll edit it in the guide.
So, without further ado, here is my guide for Ryze, The Tanky Mana *****. This guide is meant to maximize his mana, CDR, and survivability in any situation.

05/06 Accidentally hit the publish button. Later went back and finished the guide :)
05/10 Finally finished all the pictures. Thank you so much ShadowSprite and ShadowReign757 for the help :) Also added a few new runes and edited the rotation circuit. Thank you Ace13579 if you happen to read this. :)

Pros:
-Practically a spell machine gun
-Pusher Extraordinaire (via his ult)
-2.1 sec CD nuke (Q) , fully reduced (Considering that you do not cast any other spells)
-Passive reduces all other spell's CD upon casting a spell (Key to spell-mschine-gun-Ryze)
-Tanky (Excellent Health Regen, Decent Armor and Magic Resistance)
-W does a quite a bit of damage too
-Q and W scale with mana
Cons::
-Are there any? :P
-Horribly squishy and easy to harass early game
-E and R are meh
-Combo reliant
-W and E are on 8.65 sec CD and ult on 30 sec on ult, reduced fully
-Ryze's burst destroyed by Magic Resistance
-CC destroys him
-Slow
-Eh I give up *shrug*

Ryze is a fairly unique champion in LoL now that Riot revamped him, making Ryze a tanky mana ***** (this is, what, the THIRD time I typed this?), decreasing his AP ratios, his CD on his Q, and changing his ult slightly, while giving his Q and W mana scaling. THe passive remains the same:
Arcane Mastery- When Ryze casts a spell, all other spells have their cooldown reduced by 1 second. - This is what makes Ryze a apell machine gun. If you open with Overload, considering you cast 2 other spells, you basically gain another Overload. You can also spam Overload to get that snare off CD faster so that Ashe or whoever doesn't get mauled by a Cho'Gath or Garen.
Now let's move on to the actual abilites.
Overload - Ryze throws a charge of pure energy at an enemy unit for heavy damage and additional damage based upon Ryze's maximum mana. Ryze also gains passive cooldown reduction.
Passive: Ryze gains 2/4/6/8/10% cooldown reduction.
Active: Ryze blasts enemy unit for 30/55/80/105/130 (+0.2) plus 8% of his max mana in magic damage. -This is Ryze's main source of damage. Since this is on a3.5 second CD unreduced, you can basically spam this ability, in combonation with your passive. I grab this at levels 2, 4, 5, 7, and 9
Rune Prison - Ryze traps target enemy unit in a cage of runes, preventing them from moving and dealing damage. Also gains bonus damage based on Ryze's maximum mana.
Ryze traps an enemy unit, preventing them from moving for 1/1.25/1.5/1.75/2 seconds and dealing 60/95/130/165/200 (+0.6) plus 5% of his max mana in magic damage. - This is a snare that deals quite a bit of damage (Enough to rival overload, but hey, this is on a 14 second CD) Ryze can use this to escape an enemy or chase one (in accordance with Ghost and Flash) I grab this at level 1, but max it out second.
Reason I grab snare first:
If I grab a ~3.3 sec CD nuke, I'm ALWAYS tempted to spam it, which will perpettually destroy Ryze's mana before he even comes close to reaching 595 gold (I'll explain the significance of this number later). And Q does **** for damage at rank 1 and is a hugely inefficient last-hitter. Moving on.
Spell Flux - Ryze releases an orb of pure magical power which deals damage and bounces from the initial target up to 6 times. Targets hit have their magic resistance reduced.
Ryze unleashes an orb of magical power which bounces to enemy units or himself, up to 5 times. Each bounce deals 50/70/90/110/130 (+0.35) magic damage and reduces the target's Magic Resistance by 12/15/18/21/24 for 5 seconds.- This... is a little.... meh. All this is good for is for clearing minons (in combo with ult), reducing a champion's Magic Resist, and getting your Overload off CD. Grab this at level 3 and max it last. (I wonder why Riot didn't let this scale w/ mana.....)
Desperate Power - Ryze channels forbidden arcane power, granting him spell vamp, and causing all of his spells to deal AoE damage. In addition, Ryze gains a boost in maximum mana.
Passive: Ryze gains 75/150/225 bonus mana.
Active: Ryze becomes supercharged, gaining 15/20/25% spell vamp and causes his spells to deal 50% AoE damage for 5/6/7 seconds.- This... is also... a little... meh, for an ult. Compared to his old ult, this is just a Leeeeeetleee smidgeon of a nerf (No sarcasm implied AT ALL:P). The passive mana isn't that huge, nor is the spell vamp, and they nerfed the AoE damage by 15%. Still grsb this at 6, 11, and 16.

When I play Ryze, I use the following runes:
9x Greater Mark of Magic Penetration (Magic Pen.)- Extra damage
9x Greater Seal of Knowledge (Mana/lvl)- That extra bit of mana for your Q and W (+15 Q damage and +9 or 10 W damage.
The following Seals could also work for Ryze:
Greater Seal of Replenishment (Flat mana regen)- Helps with early-game mana problems
Greater Seal of Mana (Flat mana)- Helps with early-game mana problems and gives you and extra 5 Overload damage and 3 snare damage.
Greater Seals of Clarity (Mana regen/lvl)-Give you an extra 2 mana every second. I, personally, don't like these on Ryze, because I feel it gives unnecessary mana regen, considering he has at least 5k mana endgame, but these can still work.
EDIT I was talking to a friend about Ryze building when I thought "Hey, since Ryze is so subceptible to ranged autoattackers, maybe flat armor seals ( Greater Seal of Armor)?" And thus, the idea was born, this is now a legit Seal.
Now, I use 3 different types of Glyphs, the most numerous of which are:
Greater Glyph of Scaling Cooldown Reduction (CDR/level) which number 6 in total.
I also have 2 Greater Glyph of Cooldown Reduction(Flat CDR). With these Glyphs, you will have 7.03% CDR, which is just .03% overkill.
With CDR maxed out, my last Glyph is Greater Glyph of Knowledge which is 26 extra mana.
Lastly, I grab Greater Quintessence of Magic Penetration for extra damage, which in combonation of runes, Sorcerer's Shoes, and Spell Flux, you'll be able to ignore most magic resistance.
EDIT I was chatting with my buddy on Ryze builds when I realized that Magic Resist is harder to build up than Armor. With this being said, if you so wish, you can exchange these Quints with Greater Quintessence of Knowledge, which is an extra roughly 75 mana per Quint :)

I go 9/7/14 on masteries, focusing magic pen. in the offense tree, Health regen based on mana in Defence, and the rest in Utility.
The point in Swiftness can also be placed in Greed, Meditate, or Neutral Buff Duration (Your call) and the point in Flash should be placed in the other summoner spell of your choice if you don't use Flash (for whatever reason; if it is NOT possible for you to do that, then you have 1 more point to spend in Greed, Meditate, Neutral Buff Duration or Swiftness), but the rest is necessary.

Every good Ryze builds in an pretty unusual way, compared to other mages. He also grabsA DAMN Ghost BECAUSE RYZE IS TOO DAMN SLOW WITHOUT IT!!!!!!!!!(You: Want some sentence with that emphasis?) But seriously, get Ghost in every game, unless ther are weird synergies with 2 other spells that idk about
These next few spells are possible pairs with Ghost that Ryze can use:
Flash-A simple short-range blink that can help you sidestep skill shots, gain that extra distance or seriously juke from that 5-man gank. I prefer this over anything else.
Ignite-This is a DoT that can help you get first blood or stop that frikken Mundo or AP Yi from healing (Although I have rarely seen other AP Yis). However, I feel that its radius is just too short.
Exhaust- A slow/blind that can allow you to move closer or give you an edge over AD carries. However, like Ignite, I feel that its radius is too short.
Cleanse- Remove all CCs on you. What else can be said?
Don't bother with anything else.

By the way, wth is Promote?

Early game (1-6):
Grab a Mana Sapphire and 2 Health Potion, place your first skill in your snare and head to your lane (I prefer top or bot, but if you like mid, then go ahead, unless they have a more qualified mid).
Reason why I don't grab a [Meki Pendant] Meki Pendant:
For starters, you'll have to wait around for 10 more gold to get Tear of the Godess. (ik, not huge). Second, Tear is a very important item for Ryze, where he'll get most of his mana in am item, and since it take a LONG while for it to rev up, you'll want to rush it ASAP. Thus, the mana regen is wasted and you also skip out on 16 Q damage and 10 W damage, also.
You want to play agressive smart. (What this means is that you want to make smart decisions to harass and possibly deny the enemy team) You're not tanky yet (until you get your FH and Veil, consider yourself squishy) and don't want to unnecssarily lose health by making stupid decisions. Now, for instance, if your team's Caitlyn (who is possibly on a killing spree) is running away form a Garen with a Ghost up, then you might want to snare the Garen after you know good and well his Judgment is on CD, but don't dive a Rammus and an Ammumu alone to get the 1-bar [Katarina] who slipped past them. It's not worth it. (Hey, Rammus could be jungling). After getting 595 gold, you'll want to blue pill to your base to grab a Tear of the Goddess Tear of the Goddess if you're low on mana; if you still got an impressive mana pool, it is OK to farm some more until you get 945 gold to get a Tear and Boots Boots of Speed.
The next item you want to grab is, you guessed it, Sorcerer's Shoes Sorcerer's Shoes.(If there is a bunch (3+) crowd control abilities, you might want to wait 100 extra gold to grab Mercury's Treads Mercury's Treads.)
Mid Game (7-13?):
This is the point where everyone starts ganking and should start beginning by the time you have your Sorcerror's Shoes (or Mercury Treads). How you build your next 2 items depends on the enemy team's composition:
AD is Dominant:
Frozen Heart->Banshee's Veil
AP is Dominant and/or there is a F*ckload (5+)of CC's:
Banshee's Veil->Frozen Heart
AD and AP neither is dominant and there isn't too many CC's (<5):
Glacial Shroud->Banshee's Veil->Frozen Heart
Glacial Shroud is an excellent item for Ryze, giving you the 3rd most mana (fully upgraded) you will get from an item, armor and 15% CDR which will be added onto it by 5% fully upgraded, giving you exactly 40.03% CDR. Plus, once upgraded, it reduces the attack speed of enamy champions, so it's also a decent anti-AD carry, making it necassary to rush on an AD- heavy team. However, if the enemy team is AP dominant or just has CC upon CC, go rush Banshee's Veil, because the Magic Resistance, spellshield and health is worth grabbing over the wasted Armor, little extra mana, and CDR of a Glacial Shroud and Frozen Heart.

I'll take a break from discussing gameplay to discuss the famous rotation.

For starters, you almost always start with an Overload. By the time you cast 2 more spells, this spells should be ready to cast again, resulting in a 4-spell chain in, what, 1.5 seconds.
Your next spell depends on whether your enemy is running or not. If he is running, put up your Ghost and hit him with Rune Prison, followed by another Overload ( because it would be off CD by then), followed by more running with a Spell Flux integrated in and Overloads and Rune Prisons whenever possible, and DP after your second Overload.
If your target is not running away, go ahead and throw in a Spell Flux before a Rune Prison, and assume to follow the same pattern mentioned earlier. Why? DAMAGE! Your E reduces the MR of all damaged units, if you remember.

So, your rotation should be somewhat like this:
Overload-(if not running out of range)->Spell Flux->Rune Prison->Overload->DP->Overload
\ \_________
\ \
(if running out of range)-> Ghost(and Flash if needed)->Rune Prison->Spell Flux

Yep, this section is so extensive that I decided that it deserves its own chapter.
Here you should be getting your last 2 necessary items (Archangel's Staff and Rod of Ages) and the situational 6th item, which is, well, situational. Here are the options to choose from:
Rabadon's Deathcap- (You: BUT HIS AP SCALING SUCKS!!!!!!!1 N0000000000000000B!!!!!1 DOWNVOTE!!!!!!!1 ) Hang on!Don't ....... ugh. That was a JOKE! I know his AP scaling sucks! Now vote this up twice before I come over there and stuff your head in a turkey, troll!
Rod of Ages or Guardian Angel- if everything is going pretty normal, neither AD nor AP is dominant, and you're not really getting fed and nobody's stacking magic resist. Rod if you want to do more damage and GA if you really want to be tanky.
Zhonya's Hourglass or Thornmail- If you're up against an AD dominant team. Take Zhonya's for less serious AD and Thornmail if their team is composed almost entirely of AD carries (i.e. A team composed of Yi, MF, GP, Cait, and Rammus)
Abyssal Mask or Force of Nature- when against an AP dominant team. Grab Abyss for less serious AP and slow mages and FoN for serious AP, fast casters (i.e an AP Yi) or Karthus (The Embodiment of Cheap-o-rific). Abyssal Scepter can also come in handy when everybody's gets a little bit of MR
Quicksilver Sash- For SERIOUS debuffs and CCs
Void Staff-If the enemy team is getting smart and stacking magic resist
-These items I'm on the fence about, because they are kinda iffy:
Will of the Ancients- Aura AP and spell vamp. Maybe worth it if you have a Vlad on your team (Yes, it does stack)
Rylai's Crystal Scepter-AP, health and a slow with your spells. Could be worth it while chasing.
Deathfire Grasp- Gives AP, mana regen, and something to cast while waiting for your Q to come off CD. CDR would be wasted unless you switched CDR and CDR/lvl glyphs with mana/lvl. (Then it wouldn't be a complete waste...)
DO NOT GRAB ITEMS THAT FALL INTO THESE CATEGORIES!!!!!:
Items that involve autoattacking (i.e. Lich Bane, Manamune, etc.)-Ryze has way too many spells to cast.
Items that intend to max AP (i.e. Rabadon's or stacking Archangel's)- His ratios suck.
Items that actually enhance your attack damage, attack speed, critical strike, or lifesteal-I would assume you guys aren't this stupid, but I can't be too safe, and I wanted to put a third thing down to NOT get. It's just the way I roll.

Thank you for reading my guide, if you have been.
Please vote this up if you like it when you can
If you don't vote up pls leave a comment why.
Sry for the unfinished guide earlier. :)