Hi!
I've been playing Gragas since I am playing LoL and now I decided to make a guide.
This character is the one I've been playing more than any other and tried different possible builds and strategies.
My Win/Loss ratio with Gragas is very high, although I have no stats on that.

I can't code good so this will be written simply but it will contain what you need to improve your gameplay!

I make this build to help you all be better Gragas players!

Red - 9 x Magic Pen: there's nothing else that benefits AP Gragas more than this.
Yellow - 9 x Mana Regen: You will be able to spam your barrels more often early game and with 21/9/0 masteries it's very good. 9 x Health / lvl: Take those if you run 21/0/9 masteries for extra sustain although I've tried both and I prefer the Mana Regen.
Blue - 9 x AP / Lvl
Quintessences - AP

I run these masteries because I like to have that extra magic resistance / hp for extra sustain while I can use Mana Regen seals early which allows me to throw a few extra barrels (that can give you creeps/kills)

You can also run 21/0/9 for longer buff duration but it doesn't suit me, I like 21/9/0 best on most AP carries.

Start off with boots (later these are Sorceress) + 3 pots, get 1 or 2 dorans and Hextech Revolver as soon as possible it will help you a lot in lane, that extra ability power and spell vamp is beast.
Next off grab Needlessly Large Rod but if you don't have the money yet, finish your
Will of the Ancients and grab a Doran's Ring if possible.
Next up is Rabadon's Deathcap.
Now you can choose what you want to build and that depends on your team and your enemies team.
I usually build Void Staff / Abyssal Scepter and Lich Bane + 1 defensive item I need like Banshee's Veil / Guardian Angel, depending on the situation.

These setup allows me to be very effective and help my teammates with Abyssal Scepter.
For any of you who tries to tell me that Rod of Ages is better, I'd say no.

There is only 1 exception you would use Catalyst early. If you get Ahri against you.
Ahri should be able to damage you a lot, so if you want to end up well in that play Catalyst before every main item.

Simple, Max your Barrel and get 1 point in your W early then Max your E and you're good.

Flash Ignite.
I don't see anything else effective on gragas.
Flash helps you a lot, your ulti sometimes requires usage of flash.

Toss your barrel in between Caster and Melee creeps to hit them all when you want to kill fast.
DON'T PUSH THE LANE. Use your barrel so it hits the opposing champion and the caster minions, that should be best. Never push, don't be scared to go forward to last hit some of the creeps back there, if the champion hits you punish him with your abilities right after.

200 Creeps in 25 minutes should be good, if the game is very fight based you should be around 150 - 170.
If you're over that, congratulations, you're 1 step closer to play Gragas reeally well.

Any questions welcome, I don't like writing everything so I will add Q&A in here to make the guide even better.