Not Updated For Current Season
Ryze is a really easy to play champion because none of his abilities are skillshots, he has loads of mana so new (and even veteran) players can spam their abilities mid/late game and it's pretty rare for ryze to run out of mana and also he's easy because best mana items in the game are quite tanky so he can dish out lots of damage and still stay alive.
I take magic pen reds because those are the best marks for any mage.
Mana per5 seals cause it helps you spam your skills a little more early game to harrass and farm.
Cooldown/lvl blues, cause with frozen heart and runes ryze can spam abilities almost like a machine gun.
Quintessences i take flat hp cause it really helps u lane and trade blows.
If you are playing ranked and u see that you'll go up against an ad carry you can also take flat armor seals cause they will help you be more beefy and maybe even get the first blood.
If you don't want or need to be a machine gun you could switch your glyphs for magic resist/lvl to help you a little more late game.
As for quintessences, if your confident in yourself and u know that you won't need health that much then you can pick magic penetration quints as well to hit much harder during laning phase and even late game against those glass canons who don't bother to get any magic resist but go pure damage.
I go 9/7/14 because i pick up archaic knowledge in offense which in my opinion is a must have for any champion that relies on pure magic damage, except maybe if you're an amumu or some other tank.
I pick up strength of spirits cause i think that it works like a boss with ryze all game long. With this mastery you can even lane much longer cause when you pick up sapphire crystal you'll have even more mana to use with this mastery.
And i spend the rest of my points in utility picking up mana regen, xp, buffs and improve flash.
If someone wounder why i take good hands read this -> http://www.leagueoflegends.com/board/showthread.php?t=219077
Some of you might think that expanded mind is a good ability for ryze and i don't really know if it's buffed or not, cause in lol client it shows that it increases from your max mana but here it shows that it increases from base mana. Let's say that it got buffed and it increases your max mana, well i still think that it's a pretty bad mastery cause even if you get 4000 mana during the game which you will probably not, even then you only get only 200 bonus mana. I personally think that it's a waste of 4 points and i better grab buffs and flash mastery.
I start with sapphire crystal and 2 health pots (1 health, 1 mana pot if i am confident that i won't die during laning phase). I've seen some people get meki pendant and i think it's not worth it, cause even if you'll be able to lane a little longer, you'll hit for less damage and your strength of spirits will regen less health so if i think that i'll have mana problems i better take 1 mana pot over that health pot or just spam my skills a little more thinking.
After first back u want to get your tear of the goddess and first boots. With tear you'll be able to start farming mana which you'll be doing for the rest of the game and you'll hit harder as well.
Then if someone comes to gank me too often i pick up a banshee veil or glacial shroud to counter ether magic or physical damage. If i am doing really good and i don't have any problems, i pick up rod of ages cause it gives me everything i need and it stacks with time (but get this before your banshee or glacial only if you're doing good and you don't have any problems with staying alive).
For the next item i usually pick rod of ages if i haven't already bought it yet so it can do it's passive thing and start to stack up.
Then i pick up the other survivability item (banshee or glacial) depending on that, which i picked up first.
Then if i run out of health too fast and have to go back to the base all of the time i'll pick up hextech revolver, if not i'll just upgrade my glacial into a frozen heart.
Then if they are building tons of magic resist and you can't do them anything you'll want to buy a void staff, if not u can either upgrade your hextech or tear to get more spellvamp or
to hit harder.
If you still haven't finished the game and you don't need any more survivability and they don't stack magic resist in which case u won't buy void stuff, you can always pick up a rabadon's deathcup so you can hit like a boss, ace them and win the game.
If you have all your items and you still haven't finished just buy all elixirs (blue > red > green).
When Ryze casts a spell, all other spells have their cooldown reduced by 1 second.
I like this passive as it allows you to cast your spells like a mad man and use your ultimate to farm minion waves.
Ryze blasts enemy unit for 30 / 55 / 80 / 105 / 130 (+20% of ability power) plus 8% of his max mana in magic damage.
Your main damage spell, nothing much to say, nice damage, low cooldown, using with your passive allows you to use other spells faster.
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 (+60% of ability power) magic damage plus 5% of Ryze's maximum mana.
Your only cc spell, a decent snare which you'll be using to catch escaping enemies, to escape from ganks, or just to get cooldown off of your q to finish an enemy.
Ryze unleashes a bouncing orb of magical power which bounces to enemy units or himself, up to 5 times. Each bounce deals 50 / 70 / 90 / 110 / 130 (+35% of ability power) magic damage and reduces the target's Magic Resistance by 15.
Pretty bad spell, you'll be using it to either reduce someones magic resist, to activate your passive or to farm minions.
(Passive): Ryze gains 75 / 150 / 225 bonus mana.
(Active): Ryze becomes supercharged, gaining 15% / 20% / 25% spell vamp and causing his spells to deal 50% AoE damage.
Not the best ultimate but it's decent. You'll be using it to add to your burst damage as it increases your mana, to activate your passive and get cd off of your q of course, to farm minion waves, or to get back health as it gives you spell vamp as well.
I take ignite/flash, cause with ignite you can get first blood and it helps against champions like swain or mundo. Flash cause u have no escape mechanism other than your one person snare.
Also viable could be teleport, ghost and maybe exhaust.
I don't think that any other summoner will be very useful, but if you like, you can always adapt them to your play style.
Ryze is a pretty rewarding champion and very easy to master, so i think that he's good for both novices and veterans.
* I apologize for my bad english as it's not my main language.
** I also apologize for the fact that i don't have any pictures cause i have no idea how to put them in.