I use molybdenum grease from the auto parts store. A lot cheaper than what's essentially the same stuff at the bike shop.
If you aren't really into bikes, I would would just clean it all up, regrease everything and put it back together. If you do convert it to a fixed gear, I would suggest keeping all the old parts and not making any changes to the frame so you can go back to a geared bike if the fixie experiment fails.