"varisity" is one of the lowest end schwinn bikes.. you can do a lot better as a starting point. price is ok but not a bargain
as far as what parts you need... time to start googling...
it looks like a one piece crank on both so you will need to get a new crank and bottom bracket along with the right rear hub/wheel.
these days you can buy a fixed gear off the shelf for close to the same amount it will cost you to convert an old one..