Back in the early days, with albums like kerplunk and before, he used a DW kit. DW stands for Drum Works, and since then he has upgraded to much more high quality brands. Around Nimrod he got a Slingerland kit, his own custom kit and he called it the Spit Fire. Despite the popular beliefe, he didnt use this kit on the American Idiot tour.
In some drum magazine I have he said he now uses a classical kit made by ludwig, and he used variations of it in all the music videos. He uses Zildjian cymbals, and he has his own signature sticks and stick bag. He was also featured on the box of the Zildjian Rock cymbals. He still uses DW hardware and prefers Remo Heads.
His all time favorite drum is his 600 dollar snare he got before dookie, a Noble & Cooley. He likes the way it sounds.
http://www.gibson.com/whatsnew/pressrelease/1998/Jul7a.html
there is an article on the Spitfire, for those of you with slow internet connections, I'll copy and paste it.
To celebrate its 70th anniversary, Slingerland Drum Company will debut it's Tre Cool Signature Spitfire drum kit at Summer NAMM '98, the semi-annual trade show of music merchants to be held in Nashville Friday through Sunday, July 10 - 11. "Slingerland-they really make the best drums," the drummer for multiplatinum hitmakers Green Day said. "They're making a Tre Cool signature kit for me, which I'm totally psyched about. It's gonna be called the Spitfire. . . the drums will sound great."
"The main thing is that they're inexpensive ($949 suggested retail price)," he said. I wouldn't have done it at all if a set didn't sell for under $1,000. I'm not gonna tell a kid to go out and play some over-priced stuff. I can't hang with that," he said. "That's my main interest-a drum kit that kids can afford to buy."
Manufactured to Tre Cool's specifications, the Tre Cool Signature Series Spitfire kit features an 11x14 mounted tom, a 16x16 floor tom, and a 16x22 bass drum, all finished in a unique, custom "Cool" green finish, along with a 6 1/2x14 chrome snare. All kits come complete with Tre Cool Signature drumheads, a custom black Tre Cool Signature logo bass drum head, as well as a heavy-duty, chain-drive bass drum pedal, and doubled-braced snare, cymbal and hi-hat stands. Six-ply shells with heavy-duty tom mounts, bass drum spurs, and floor to legs make the Slingerland Tre Cool Signature Spitfire kit rugged and durable.
He uses 16 x 16 and 16 x 18 toms. 14 x 11 rack toms, and a 14 x 6.5 snare and an 18 x 22 bass drum. He also has an assortment of toys.
Aside from drums, Tre plays other things. I couldnt find what accordian he uses on Warning and I don't know what kind of guita he uses in his solos, but Im guessing one of billies.
http://www.zildjian.com/EN-US/products/configurator/default.ad2?artistID=2638
there is a view of his kit.