OK, my opinion. NOT GOOD to game/internet/etc..... on your recording rig. I have heard so many people complain about unstable PC based rigs, 9 out of ten of them are using thier all purpose computer for their recording rig. I built a machine specificly for recording, no non-recording related applications on it, and I have yet for it to ever crash or suffer a driver conflict. It does not hiccup, pop, etc....... I will put it up against a Mac any day.
About monitors, a good budget setup would be some Behringer Truth powered monitors. They run around $250 a pair on eBay new, and will fill a small room with FLAT RESPONSE sound. Of course they are tunable to compensate for room acoustic anamolies. They aint Mackie or Alesis, but they serve well in a budget monitor role.
Also, Beyer makes some really good headphones, relly good, with great low frequency response and the best isolation of of headphones this side of $500. you can get them for unde $200. A lot of people suggest not mixing with headphones, but they can be an invaluable tool.
In an ideal situation you would have a dedicated PC or Mac, good monitors, good headphones, AND a small PA system. The PA is good because you can use it both for jamming during rehearsals and to give you a good live feel while recording, then only use your monitors during mix/edit/mastering.
I am lucky, I have a small budget setup but have a good friend who owns a sweet studio with perfect rooms and an array of monitors spaced throughout the control room. So when I really need many points of reference, I can drive 5 miles up the road and use his setup.
One last thing. Nothing is going to give you a more accurate picture of what the end user hears better than a good car and home stereo. After you have achieved a good mix, it is always important to hear it in a variety of "non studio" environments. The end user is not going to give a shit what you used to monitor your recording. I know of a studio here in Jacksonville Florida where the engineer takes bands to his car to hear the mixes, it is a cheap tactic to impress them with the quality of what he has done for them, but it keeps them coming back!!!!!