Because you want to convert your basement to a living area (rec room, family room, weight room, etc), it is usually nice to have a floor that isn't cold all of the time.
The textbook method is to put down at least 4 mil plastic sheeting to keep moisture from rising into your flooring, then build 2X4 framing, laying each on their long sides, attaching them to the concrete using concrete anchors (all wood is pressure treated) and putting 1" rigid insulation between these 2X4 sleepers. Then put 3/4" plywood flooring over this and then your carpet, HW flooring or vinyl over this. However, this method will eat up at leat 4 1/2 and up to 5 1/2" of your headroom.
Your best bet, IMO, is to put down a moisture resistant carpet that is held in place with water proof adhesive. A flooring shop or even a good "floor and wall" sales rep at Home Depot should be able to help you with this. Installing it is not that hard, but does require a little bit of mechanical skill and a few tools. If this is something that you think you would like to do then go for it. If your just doing it to save money, I'd recommend you hire it done......the results will be worth it.