I've started tanning recently, and I don't go THAT often, maybe 3 or 4 times a week. But the past 2 or 3 times I've gotten little bumps on my chest and stomach after I tan. I've been peeling there too, so I figured that could be the problem. But it itches a LOT as well, but that always happens when I get burned, but when I itch it a lot, more bumps pop up. This is normal right?

well, the bumps, if you are peeling, could very well be tiny blisters from your skin being damaged.

Or, if you are using oils during this process, perhaps pimples?

Those are my guesses.

after tanning for 2 years in a row- my guess is you are tanning to quickly. If you are burning then you need to cut back your time or go to a bed with less wattage.

build it up over a few weeks, the day of and/or after you tan you should be a light pink not red and have no pain!

Go no more than 3 times each week until you build a tan and then cut back to once or twice to maintain it. Use lots of moiturizer after showering and after tanning to retain healthier skin.
I agree with Sher:
the bumps sound like blisters from burning, the sweat builiding up under the damaged skin, then it peels off. If you aren't careful you will get "sun spots" and be all blotchy. I used to use self tanning stuff on the off days so that I didn't have to go as often to the tanning bed.

Actually it is not normal. Your skin is trying to let you know that the amount of time you are spending "in the sun" is too damaging to your skin. It sounds like you are having a skin rash allergic reaction, could be from the heat, could be from to much UV light, could even be from the tanning oils you use. It even could be sweat building up under the skin and appearing as water blisters. Both Sher and Shawn are offering you good advice!

P.S. Ever hear of self tanner? Toma's Self Tanner is the BEST!