Well, I'm probably about to get blasted but here's what I'll throw out there. Keep in mind, I'm a Nikon guy so I'm not too keen on the Canon stuff.
Tripod= $80 on sale at
Best Buy I have this one and like it for the money. The reason I bought it was to shoot fireworks on Main Street. I've used it sooo much for so many different things since than. The middle also comes out and acts like a monopod. What I don't like about it- It's big and heavy compared to just about every other tripod. Also, the head would unscrew from the monopod but a dab of gorilla glue a year ago fixed that. (I think my status just dropped even lower with that statement

) It may not be worth a tank of heating oil but it holds my camera nice and steady even with my Sigma 18-250 on it which weighs 1.4 lbs without the camera body.
Flash=$100 used at B&H. Before you do the look like you just tasted the Beverly in Club Cool at Epcot, think about it for a second. I have the same flash for my Nikon and I can take a flash picture from over 100 feet away. Granted it's used but it has a rating of 9 out of 10. When I bought my Sigma lens from them it was used with a rating of 8 and it looked brand spanking new.
55-200 Lens= $160 from B&H. It's a Sigma but made for Canon. It's not an expensive fast lens, but it's under budget and an option. I can't attest to this one in particular but I like the Sigma brand. Do your homework and read some reviews. Don't buy it because the link is here and it's easy
Bag= $80. Bags are all personal preference. Go to a store to hold a few in your hands and try them on. I have the older discontinued version of this one. I literally needed one in a few minutes and ran into Best Buy on the way to the shore. It was there, I grabbed it and have been EXTREMELY happy with it for what I do. It stays packed and by the front door so I can grab it and run without thinking.
Battery =$45 Ever walk around Disney enjoying the sweet sweet sound of every shutter actuation and see your battery warning light go red? Ever try to tell yourself you didn't
need to take a picture of that? Get an extra battery and keep it charged.
Whatever is left over, I would get a decent 30/mbs memory card probably 16GB (I was in Staples yesterday and they had some Sandisk extremes on sale for about $30) and probably a remote shutter release. For the remote, I don't think the T3 has an infrared port, so you may need a
wired remote which some people prefer anyway.
So, that's my thought process. Is it the best stuff? No, but it's worked out for me so far I think.
Edit: Go an extra $20 over budget and get a comfortable strap too! Nothing like a red neck in the summer from the stock camera strap digging in!