OK this is going to be a bit long because polishing is more than just polishing...
I do polishing/paint correction for family and friends as more of hobby than anything else, but I have done show cars for Sema and other shows in the past to make some side cash.
Definitely at least get the Porter Cable polisher like someone else suggested. It's a good mid-level polisher that will get the job done and is fairly easy to use. I personally use a Cyclo Dual Orbital and love it.
I'd recommend washing the car with Dawn dish soap to strip off any existing waxes and/or sealants.
After that, use a clay bar to remove and contaminants that might be embedded in the paint. If you don't feel like doing the whole car (could take an hour or so), just hit the major areas susceptible to contaminants (front bumper, hood, front fenders and at panels behind wheels). You can pick up a cheap Meguiars clay bar at Auto Zone or O'Reilly for around $5 or so. With a latex glove on (or even bare handed) rip it in half, spray it with your favorite quick detailer to get it nice and slick, and start going to town on the areas you want to, while spraying the areas your working on with the quick detailer. Make sure to keep the surface slick. You'll be amazed at how much gunk you'll pick up with the clay bar. After you're done, you want to wash the car again, this time with your regular car, wash soap (I personally love Poorboys Super Slick n Suds
Super Slick & Suds Car Wash - Poorboy's World Inc.
After washing, break out the Porter Cable, throw on a white or blue pad (lighter cutting), out a few drops of a light cutting polishing compound (something like Meguiars M205
Meguiar's Ultra Finishing Polish M205 - 8 oz | Free Shipping Available - Detailed Image) and start off with a small area. Work slow, moving the pad an inch or so per second with little pressure. After you're done with a 2'x2' area, shut it off and wipe it down with a microfiber towel to see what you've got going on. Use a good halogen lamp or even an LED flashlight to look for imperfections still showing through the paint. If you see good enough results just using the light cutting compound, by all means, move forward with it around the entire car using the same method. You'll likely use a few pads and a decent amount of microfiber towels (they get saturated with product) so be sure to stock up.
If you notice a lot of imperfections, slap on a yellow pad and use Meguiars M105, which is a deeper cutting compound. Use the same method. Know that of you do this, you'll have to follow it with M205 after you're done as the paint will have a hazing to it.
Once you're done, you can apply what ever carnauba based wax you desire. I do most of my wax application by hand, but some like to use the Porter Cable too.
After you're done, you should have a dam good looking car!
Note - I usually spend anywhere from 10 to 30 hours on one car, so be sure to be patient when doing this. It'll all be worth it in the end!
