So. At first I want to introduce Gragas.. In my opinion, Gragas is the most effective Pusher oO I know.. You'll think it's rubbish. But you can try it out and wondering :)
So let's start.

Pros
- Great Farmer
- Great Pusher
- Early Game Domination
- This damage...
- unkillable? O:

Cons
- Early Game mana problems
- Maybe focussed cuz of Mejai's
- Ulti needs a bit practice
- Only skillshots

I want to start with the runes.
I use 3x They give you enough of the AP, which you need for some early-games-ganks
9x Just to get magic penetration. In my build, I don't use some items with penetration. so this is the only MP, which you get.
9x Gragas has to fight with his manabar >_> and your isn't just enought mana-regen for you. So! Choose this runes.
9x What should I say. Your needs to get a low CD, so I choose this runetype.

I choose some 9-21-0
I dunno what I should say 'bout that
I mean.. First 9 points for more AP/CDR and magicpen
The 21 in the second tree, cuz we don't need any manathings or something from the last tree.
Here you'll have more HP and AP and ofc the damage reduce, which is never false

Well. The Items
I always start with a . You'll have enough AP/Manaregen/HP with him. The perfect item to start with.
I go on with a Catalyst the Protector to get a little HP/Mana-boost. You will need it.. And the passive isn't as bad as much player think, imo. Always nice in some earlygame-teamfights, when you'll level up and the enemies are like 'ughugh.. oh noez! shizzle!'
To get a little more speed, I collect some money for . Ok. Shoes..
Imo they are the best choose for Gragas.. BUT always buy the shoes, which you think, are right for the situation.. You know what I mean.
Next Item will be . If you see, that the enemies are to strong.. oh gawd.. please don't buy it. Nonsense and so on..
Time to get this . Mid game has started. You need time to get the whole stats from it.
So now to get the shiat a bit tanky. will be the right choice. 'nuff to say about it. Nice HPboost, Nice APboost and at last a nice passive.. So the enemy can't flee. NEVER.
Maybe if you're the perfect Gragas midgame continues and you're like a boss.. Then get some . Yeah.. 'nuff to say about it. Just the most powerfull AP boost. To end the build, I always choose . Just to be like a fleshshield and with combination with your nothing can stop you :)
Don't forget to buy some potions at the end or some oracle if you need it.

Like many other champions, Gragas hasn't any 'real' skill sequence.
Let me describe a few combos :)

At first. Always.. Realy always when it's available, use . It costs nothing and gives you a realy strong buff. And ofc a little bit mana.

Next. Some & -Harrass. Ok.. let's place your barrel in front of the direction, in which the enemy-champ runs, then do your E on him. Exact in the same moment let your barrel explode. Great damage :)

You can combine this with your ultime too, to get a closer distance to realy far aways champs.
The next combo is only for fleeing and I use it only, if I have no chance.
& & perhaps and/or
No one can follow you.. too much blinks.. too much speed and so on.

That are the combos, which I mainly use. Don't forget, that you can throw the enemy through walls with :)

Yeah.. ok.. My choice is and
Need it for runnin' away :)
If you want a more aggressive Gragas, try instead of the Flash.
If you have some question about the other summonerspells, plz ask. I'm only to lazy to write all the others and a 'Why not?'. k

So far.. It was my first guide for a champion which is played so rarely.
Please give me some adives, what I can do better.
And.. plx no flames about my english ._.
I'm only a german*** and I did my best.
Hope you have a few nice games with it :)
Bye