The first time I used Gragas I followed this guide and I haven't changed yet:
The Drunken Master
The basic idea behind this build is to stack health and cooldown reductions. Gragas' passive
Happy Hour grants 2% of his HP over 4 seconds everytime he uses an ability. His skills are pretty spammable and have uses both defensive and offensive.
His
Barrel Roll is a long range aoe remote mine. It will detonate after 5 seconds, or you can detonate it yourself by pressing it again. You can launch this through trees and everything.
Drunken Rage is a no cost 30 sec cooldown that gives a 20 sec buff on AD, lowers dmg taken and gives you 50-90 mana. Unbelievable considering it also gives 2% HP regen from Happy Hour.
Body Slam is a charge that aoes on impact, damage and slowing enemies. Short cooldown, even shorter late game. Valuable skill for escapes, quick movement around the map, but most of all chases/tactical positioning.
Explosive Cask is a nuke that blasts enemies away from the target area. Decent dmg, long range, very useful for saving allies or repositioning enemies. Also lowers enemy attack speed.
tl;dr Gragas is a deceptively agile and manipulative hero with decent dmg and massive survivability through self-regen.