If you are a beginner in detailing then i'll recommend that you let the experts do the work for you. You described the paint as dull and having lots of swirls and scratches. To remove this problem you'll need to use a Polishing Compound to remove all/most of the crap on your paint's clearcoat. This will give excellent results when done with a High Speed Rotary OR an Orbital Buffer which is way better than doing it by hand. It is important to keep the buffer moving at all times and not to apply too much pressure as it can 'burn' or remove an excessive layer of paint; you don't want that to happen. A polish will remove overspray from your paint.
NOTE:
Polishes and Waxes are two different things.
A polish removes the dirt, grime, oxidation and a part of the paint for restoring the shine of your car. It is an abrasive and does not give any protection to your paint but only 'cleans' it.
A wax protects the paint from factors like sunlight, environment pollution etc , fills in minor imperfections (swirls) and gives a great shine. Wax is applied by hand in a circular motion, evenly on the paint and then allowed to dry to a haze. Then the wax residue is cleaned out preferably with a microfiber towel. It is very important to wax after polishing as a polish also removes previous wax.
My point is, have your car polished and waxed by experts then look after it yourself (wax it regularly and also use clay twice a year) as to prevent future problems.
*Also see the detailing notes by @vague_one
