Hi! I hear you want to learn how to play Ryze?
This guide will document my personal experience with Ryze, with countless games of experience.
Ryze is a unique champion in League as he scales primarily off maximum mana, as well as ability power. This makes Ryze a bit more tricky to build than a standard caster, but also allows him to build a huge array of effective items.
Ryze is generally played in mid lane as high damage mage that has huge survivability. He is an effective pick against almost all AP carries and should beat them in lane after the core item build is complete. Ryze is completely item dependent, but he scales into the late game better than any other mage.
Please keep in mind while reading this guide that this is just the way I play Ryze, the best way to become godly with a character is to experiment and find a build that suits your play style.

P.S. This may seem like a lot of text at some points, but if find text is much easier to understand than using icons every time I want to reference an item/skill.

P.P.S PLEASE GIVE FEEDBACK <3

Ryze is easily picked up, yet offers a lot to his team. Basic Ryze play can be very simple, but complex skill combo's make him a high skillcap character, despite his lack of skillshots. Ryze dumps on other AP mids, and carries a game almost as well as Kassadin and LeBlanc. In a long game, Ryze the best caster to have on your team.
Ryze becomes really tanky through items and has so much room to make mistakes and still live. Ryze is a common pick in competitive play, due to his consistency and ability to carry games.
He is also a good pick for higher ping players (200+), as he has no skillshots.
If you take the time to master Ryze, you will gain ELO.

As Ryze you have 3 stats that increase your damage:
Magic Pen, Max Mana, Ability Power.
Many Ryze players forget that ALL YOUR ITEMS DO NOT NEED TO BE MANA ITEMS
Dont feel bad if you are called a noob by another player for building a Rylais or Abyssal Scepter, they dont know what they are talking about.

I thoroughly recommend smartcasting on Ryze as you find clicking on targets will slow your rotation and will result in less damage.

Spellvamp works best on champions that have a consistent rotation of skills that allow them to consistently replenish health. Rumble, for example, has incredibly low cooldowns and will be able to regen a lot of health over time, while a burst mage like Leblanc will only heal during her burst combo.
Spellvamp is a necessity on Ryze. And the best way to get spellvamp is through the item "Will of the Ancients". WOTA provides a passive spellvamp and ability power that stacks with every aditional WOTA, allowing you to gain crazy ammounts of health with your skills.

First off, boots and pots, against any lane.
Boots can be built into Mercury Treads or Sorcerers Shoes, depending on personal preference. (I usually go with sorcs).

This build revolves around building multiple items at once. This tactic maximizes mid game advantage by filling all inventory spots and efficent use of gold. The 4 items that are considered "core" that should be built EVERY GAME along with boots are:
>Tear of the Godess
>Catalyst the Protector
>Will of the Ancients
>Glacial Shroud

Catalyst and Shroud can further be upgraded to a Banshees Veil and a Frozen Heart, the two items that provide you with mana and survivability.

After these two items, you may craft your tear into an Archangels Staff, before deciding upon a 6th item. Common selections are:
Void Staff for increased overall damage.
Archangels Staff for more damage and magic resist.
Rylais Crystal Scepter for more health and slow.
Rabadons Deathcap for more damage (not personally recomended)

Other items that work on anyone:
Guardians Angel, amazing if you accidently get caught.
Shurelya's Reverie every team needs one of these.
Aeigis of Legion every team needs one of these.
Quick Silver Sash, are you being killed by Warwick or Malzahar?

This 6th item, as well as the way you build your core is completely individual. The only way you will learn what you prefer is through repeated practice. As you progress you will learn to build to the needs of each game, and adapt.

Before Tear (Early laning phase)
This will be your weakest point in the game in terms of damage. If you against a mage, the magic resist blues make the difference between surviving to get your tear and giving away first blood. Remember to only use your Q skill (Overload) when you really need to, and don't bother with harassment unless you feel the minions will help your damage trade. Ryze auto attack animation is smooth, and if you practice you will find yourself keeping up, despite early pressure.

Building Core Items (Later laning phase)
As you progress towards having the 4 core items (tear, boots, catalyst, revolver) you will find you are gradually growing stronger. You may now begin to harass with the combo Q>W>E>Q or just throw out random Q's as your opponent goes to last hit creeps. If you feel like you can get a kill, be sure to use your ult and ignite/flash if necessary. If you manage to push your opponent out of lane, remember that ganking other lanes is key to winning a match. Dont be afraid to flash in for a W stun to allow your team mates to land their abilities and secure a kill. If you have a nice jungler you will be fed blue, thank him and proceed to spam abilities.

After Core Items(Midgame)
You will be strong, but not incredibly tanky. Roam with your team, start buying blue pots whenever you can afford them. Blue pot, Frozen Heart and level 5 Q will max your cooldowns, keep this in mind. Max cd = Max damage. Remember to push out lanes by farming the creeps that are close to your base and try to obtain objectives as a team.

Big Items Complete (Lategame)
After you build a Frozen Heart, Banshees Veil and a WOTA, you reach a whole new level. By this stage you will be really strong. Now is your chance to be super aggressive. Dont be afraid to jump right into team fights, unlike other mages, just remember you are not the tank. A Q>W>E>R combo with your ult up should melt anyone with no magic resist and is really effective at destroying overextended squishies. The longer a fight drags out, the stronger you become, it is not uncommon for you to be able to ult twice in a long fight, providing an extra burst in damage to clean up the oposition.

PENTAKILL MACHINE (45 mins and more)
If the game goes this long, and you complete an Archangels and an item of choice, you step up again. Now they have no chance. With the right 6th item, you are now a free win machine. If you have spare gold buy Elixirs and Oracles. It is not uncommon to be able to 1v3 enemys, as long as the AD carry isnt there. Just keep spamming skills and you win.

If you feel like this guide has no more to offer, then you should check out some videos of pro's playing Ryze. I recommend watching bigfatlp (bigfatjiji) or Alex Ich, two of the best Ryze players.

Bigfatlp Stream: http://www.own3d.tv/live/13576/CLG_bigfatlp
Bigfatlp Videos: http://www.youtube.com/user/GeneralWiser/search?query=ryze

Alex Ich Stream: www.own3d.tv/live/136253/M5_Alex_Ich

Here is an awesome video showing some Ryze highlights. ( I did not make this video).
http://www.youtube.com/watch?feature=player_embedded&v=JgXz-_tjdLA
Thanks Biddle :)

1.0 - Guide created 1/30/2011.
1.1 - Fixed typos 1/30/2011.