I play Gragas the same way I play most APs on Aram. Start chalice, boots, and potions and go for Athene's as your first item. You lose some damage this way, but you can spam the hell out of your spells.
For Blitz, you might start tear and try for a tanky mana build with frozen heart, sheen-->iceborn and finishing the tear (not sure which one is better, AA or MM)
Random ARAM tips:
Always get a bulwark on your team, and you should build it if you are the best person to get it on and no one else is getting it. I usually rush it in most cases, just gives a huge boost to your team if you have it and the other team doesn't.
Also please never build gp5 on anyone, I cry whenever I see people starting with like, philo, or even worse building it 5+ mins into the game. Like not only does it suck but it might not even repay itself by the time the game ends.
Same reason chalice is better than tear on a lot of champs besides those that can stack tear super fast.
I do Chalice + Boots + pots on both. I usually finish Athene's on Gragas and just sit on the chalice on Blitz and build tanky from there.
These are my major starts:
1)Chalice/Tear + Boots + pots - It's usually Chalice unless it's someone who stacks Tear quickly.
2)2x Doran's Rings + Boots + pots - Misc APs that I don't build mana items on (TF, AP Yi).
3)Boots + Kindlegem + pots - My opening for tanky champs (except Singed where I do Tear). I almost always buy Spirit of the Ancient Golem since I think that item is stupidly good, but you could also go Spirit Visage (another good item). Though often times, I find myself getting them both anyway.
What do people do on AD carries? I'm undecided between 2x Doran's or Vamp Scepter.
I usually decide on dorans vs vamp scepter depending on range and safety. Aura items in general are extremely good in Arams, If I'm a tanky champ I usually try and get as many as I can.
What are some mistakes you guys see that people might not realize they are doing? I feel the biggest one I see is people not dying when they are low HP. Always gets me annoyed when someone on my team gets very low with no sustain, and for the next few minutes plays extremely defensive making it almost a 4v5 for the duration. I would say I have won a bunch because we got people to 100hp, and then they sat in the back doing nothing while we took turrets and got picks.
not dying (prefer by execution), not taking health packs, getting poked to death without ever initiating, not having an aegis/bulwark, using flash as a summoner spell
flash is pretty bad since if you flash away from something that would have killed you, now what? it's not like you can heal up. you're just worthless sitting around at 25 hp. it is still good on some champs like fiddle, annie, etc.
I got Zac three times in a six game run last week. But I only own about 35 champions, so there's going to be a lot of repeating. I wish I got Ziggs more though, since not only is he OP in ARAM, he's my favorite champion. I think I've only gotten him once in over 100 games.