I know what you are thinking: AP Ezreal sucks! Well, you are wrong. I still play and love Ezreal with an AP build and i own the enemies. The difference to an AD Ezreal is, that you do heavy burst damage to not only one champ but the whole team at once with your ultimate and your essence flux. Of course that also includes burst damage to single champs in 1on1 fights.

Pros
1. excellent mid champ
2. high mana regenration because of runes and Archangel's Staff
3. high damage output to all champions
4. you are able to slow many champions at once with Rylai's Crystal Scepter
5. you are able to chase enemy champions down with Arcane Shift + slow

Cons

1. very squishy
2. if you don't get enough AP you are pretty useless
3. you get focused in teamfights
4. difficult to farm

Passive
Hitting a target with any of Ezreal's abilities increases his attack speed by 10% for 6 seconds, stacking up to 5 times.

Pretty good for an AD Ezreal but uninteresting for us, though it's usefull to get some last hits on champions in teamfights.

Mystic Shot
Ezreal fires a bolt of energy. If it strikes an enemy unit it reduces all of Ezreal's cooldowns by 1 second. Applies on-hit effects.

Though we don't do many damage with this skill it's still useful to get some last hits on creeps but most important is that our cooldowns get reduced by one second every time we use it. With our high mana regeneration we can spam this spell so we can for example reduce the cooldown of our ulti by about 15 seconds.

Essence Flux
Ezreal fires a wave of energy at a targeted area. Any enemy champions it passes through are damaged and have their attack speed slowed for 5 seconds, while any allied champions it passes through have their attack speed increased for 5 seconds.

That's our main spell. We do nice damage to multiple targets. It's also good against physical DPS champions because we reduce their attack speed.

Arcane Shift
Ezreal teleports to a nearby target location and fires a homing arrow at the nearest enemy unit, dealing damage to it.

This spell actually does the most damage in the end when we got some nice AP (except ultimate and only to one target) but we normally use this spell to get to the enemies and start the E-W-Q-combo or we use it to run away. Together with Flash you cover a nice distance in a short time.

Trueshot Barrage
Ezreal channels for 1 second to fire a powerful barrage of energy missiles, which deal damage to each enemy unit they pass through. However, it deals 8% less damage for each unit it hits with a limit of 30% less damage for any hit after the first 4.

I love this ulti and it's and you do devastating damage to multiple champions with this build. In late game i normally do over 1000 damage with this ability. Imagine for example that you see all your enemies pushing the mid lane and before the teamfight starts you cast you ultimate that hits all your enemies. Then it's yout team with full health against the enemy team with everyone at 50%(squishies) - 80%(tanks) health. That gives your team a huge advantage.

Flash:

Good to chase down enemy champions. You can flash through walls and it's awesome in combination with Arcane Shift. Instead of Flash you can use Ghost.

Ignite:

I use it to get an easy first blood or to reduce the healing of champions like Dr. Mundo, Vladimir or simply champs with much life steal/spell vamp. It also does a nice damage at higher levels.

We start with a Meki Pendant and two health potions so you shouldn't have too much mana problems. If you are low on mana there is no reason to go back to base because you just have to wait a little to get your mana high again. When you have enough gold, grab yourself some boots and a Tear of the Goddess to start increasing your mana, so you have a nice AP boost with Archangel's Staff. If you don't have enough money for the Staff and Sorcerer's Shoes the next time you are a the shop than i would first buy the shoes first because Ezreal is quite slow and that's bad for a squishy like him.
If we die often and/or don't get many kills then don't buy Mejai's soulstealer or at least wait until it's getting better.
The last thing i can say to the item build is that you can buy Rylai's Crystal Scepter before Rabandon's Deathcap so you get the HP boost and the slow earlier but that depends on how we are doing in the game.

EDIT: Instead of Void Staff, Lichbane is also a great choice for Ezreal because it adds 100% of your AP to your Mystic Shot so you also do great damage with this skill.
When you choose between one of this items you have to consider if you have many champions in the enemy team with high magic resistance. If so, i recommend to buy void staff but if not, your maximum damage output might be increased if you buy Lichbane.

The best lane for us is the mid lane but if one of your team wants to go mid, then that's not a big problem.
We try to last hit minions with our Q-skill and also try to hit the enemy. The advange of our W-spell-focus is that we don't have to care about minions between us and the enemy and when this spell reached level 3 we do very nice damage with it. Harass the enemy with Essence Flux constantly and the HP will drop fast. After some hits the enemy will try to stay out of range but with a suprising E-W-combo you can still hit them. Finish them off with ignite and melee attacks or your ultimate when you reached level 6. Also watch the other lanes and use your ultimate if an enemy is low and wants port back to base. That's an easy kill and it gives you some extra money and an advantage against the enemy on your own lane.
If you did well you can kill squishies in just a few seconds and you do a nice damage to multiple targets in team fights. Always stay behind your teammates because you die like nothing and try to hit as much enemies as possible with our ultimate. One last thing: Blue Buff is your best friend.

17.09.11
- i changed the skill sequence, so essence flux gets maxed as soon as possible, because this makes it very easy to harras the enemy champ with this spell
- i changed my masteries a little bit. i took 4 points away from expanded mind to max awarness and utility mastery, because when you have 2000 mana you would only get like 100 mana with expanded mind.
- i replaced the greater seals of replenishment with greater seals of clarity, so i have 9 greater seals of clarity instead of 5 because when you are lvl 6, these seals give you the same mana reg and at that point, you normally dont already have an archangel's staff. i also replaced the greater quintessence of intellect with a greater quintessence of swiftness, because i think the extra little movement speed is more usefull than 38 more mana.

That's my first guide, so I am not very experienced.
I only show you how I play and I do quite good with Ezreal. AD Ezreal also works fine but I prefer this build.