You hook the speakers up to the audio-out jacks on the TV, assuming it has some (a 51" ought to). Hook your source components up to the TV directly. The subwoofers connection would depend on the available audio outputs on the TV. Btw, another advantage of this setup is that the speakers are higher quality ones than if you had to dilute them by making room in your budget for an AVR.
Another good option you might have currently, if there is a Fry's in your area, is to see if they have the Infinity Primus p363 speaker on sale at that location. Fantastic speaker, and at many Fry's it is only $200 a pair right now. Get a decent receiver, for 300, you might try used ones on ebay. If you need HDMI connectivity, a good used one is the Onkyo tx-sr707. Also look for used Onkyo tx-sr805s. If you don't need HDMI, check out some refurbished Harman/Kardon 3490 stereo receivers. Skip the subwoofer for now, maybe get one later- the Infinity speakers have pretty good bass on their own. The Infinity speakers are competitive with speakers far more expensive with respect to sound quality and performance.