Rear wheel spokes are likely to break if they were undertensioned in the first place, which on a low-end bike is a real possibility. I see you're in Nashville and Chicago. Where did you buy the bike?

This is happening because as you apply torque to the hub, the spokes are getting tensioned and detensioned, and with enough wear cycles they break. If you're not breaking only one "position" of spoke (usually left side spokes of the "leading" sort, which means they join the rim counterclockwise from where they leave the hub), it's probably a general problem with the wheel, and it should be relaced. You can tackle this yourself if you're feeling ambitious; it's not all that difficult, and sheldon has good instructions. If you're going to pay a shop to do it, I would have them look at the overall condition of the wheel and give you an honest assessment of whether it would be better to put your money into new stuff.