Buy yourself some hand glaze and some good carnuba wax. I'm partial to Meguiar's professional line stuff. I would suggest buying a buffer and some good bonnets, but this is the $50 answer, not the $150
The basic three categories are rubbing/polishing which are a little hefty for a few swirls, glazing which can be done by either hand or machine depending on the glaze you get, and then wax/protectant.
Get some hand glaze (make sure its not machine glaze) and grease up your elbows. Follow the directions. One of the reasons I prefer the Meguiars professional line is because the polishing agents (only slightly present in the glaze) are designed to break down as you use them. They start out a tiny bit more aggressive, then as you work it and it dries, the cleaners become smaller and softer. The longer you work it, the more good you're doing. Follow it up with a good liquid carnuba wax.
You don't need to start with any wax remover. The glaze is strong enough to cut out scratches in the paint, it'll cut through the wax.