If you have a lot of swirls then removing them will give you a deeper shine.
A machine polisher will give you better and faster results because it whizzes round at 500rpm and because the heat that this generates allows you to use more specialised polishing products that start off relatively abrasive and then break down into finer compounds.
By hand you have to rub away for hours using an abrasive polish and then switch to a finer one. It's all too much of a drag basically.
Use a dual-action polisher rather than a rotary .. they're safer for a novice and it's practically impossible to wreck the paint. Rotary's are more lary and you need to know what you're doing or you'll trash the paint.
Use the mildest polish that you need to get the job done. Start with the mildest there is, in the range you're using (unless you see the paint is really bad already) and see what happens. If it's taking too long to get anywhere, go to a more abrasive polish instead, then finish off with the milder one.
If you want to go all anoraky you can also vary the pad/polish combo's to give you different combinations of "cut".. but save that one for later.
There's millions of machine polishes out there. I use Poorboys as their system is easy to understand. SSR1 is mild, SSR2 is slightly less mild and SSR2.5 is a teeny bit more than less mild.. you get the idea.
You can still get good results by hand.. but choose a polish that is designed or doesn't mind being applied by hand. Autoglym Super Resin Polish is mildly abrasive and contains fillers that mask the swirls it doesn't remove (though obviously they'll appear again later on). I'd avoid T-Cut or cutting comppunds ,.. they'll be way too fierce (in Poorboys terms T-cut would be like SSR3).
The coloured polishes (Colour Magic etc) can give good results too by hand. They're less abrasive than T-Cut, a bit more abrasive than AG Super Resin Polish and are filler-heavy, so you'll get a gleaming result on red in particular.
Whatever system you use, wash teh car thoroughly before you start, clay it, polish it and then stick a wax over the top. Even the "cheapo" tins of Simoniz or Turtle Wax they sell in Halfords will do the job.. but you may want to try Autoglym HD Wax (
) or (slightly cheaper) something like AG's Extra Gloss Protection, which is a fluid that keeps the AG SRP fillers in place a bit longer than otherwise.
But you're into the realms of alchemy now.. different people will tell you all sorts of their preferences.. but the fun part is finding your own.