removing swirls is really tough by hand...REALLY tough.
Ideally you'll need a two stage correction to remove them and restore your paintwork. Stage 1 cuts the clearcoat/lacquer back to remove/reduce the swirls, stage 2 refines and polishes the clearcoat/lacquer to a high shine. This is all achieved using either a dual action or rotary polisher and appropriate pads and compounds and possibly wet sanding if some scratches require it. Most detailing companies will do this work without a problem but be warned, done properly this is not a quick or simple process and involves multiple stages.
1) the car is pre-washed/snow foamed then rinsed
2) The car is two bucket washed then rinsed.
3) Tar & glue remover is applied and rinsed
4) Iron fallout remover is applied and rinsed
5) The car is clay barred.
6) The car is again two bucket washed, rinsed and dried.
7) The cars paint is inspected using sun lamps and paint depth measurements are taken from all over the car.
8) The cars trim, lights, badges, window seals etc are taped off to avoid getting polish on them
9) two or more passes are made with a polisher using a course cutting compound, this can take several hours depending on multiple factors.
10) Two or more passes are made with a polisher using a polishing compound.
11) If necessary a glaze may be applied to cover any swirls/scratches that were too deep to be polished out (not always required)
12) The car is wiped down with IPA to remove oil residues from the polishes/compounds.
13) A sealant or wax is applied (sometimes both).
(obviously I've not mentioned cleaning the wheels)
The above will usually take 2-3 days and the price varies around the country and from detailer to detailer. As a guide when I did my GT last it took two full days and I'd have charged around £350.