How can I remove a non-load bearing wall in my basement?
You really need to be careful about removing that wall. I hope there are three rooms in the basement because this sounds load bearing. Is the wall bisecting the basement? If it is a normal non-load bearing wall, my advice is to just go for it. 

Find the circuit connected to the wires in the wall, turn it off at the breaker, and cut them out of they way. After you cut them, cap each of the wires keeping them separate and put them into a temporary junction box. 

For the Studs, the easiest way to remove them is cutting the nails at the bottom of each one with a reciprocating saw. If you don't have one, beat them out of the way. My guess is that it is old lumber and you have plenty of heavy nails to deal with so "beating them" may be back breaking. Find a friend who has one or pay the neighbor kid a few dollars an hour. 

Good luck with the carpet...too many options here. 

And save that old lumber. That is good stuff. You just cant use it with new lumber because it is too thick.