Cutting compound

[Face.]

Here's just what I'd do.

wash the car first with a car shampoo of your choice and not fairy liquid like some people do.Remove all paint contaminants.

I have never used a clay bar but have heard good things and apparently you can use blue/white tack that is wet(keep wet) as a cheap once use clay bar.

then polish keeping the car damp and away from sunshine.Little circles and try using a micro fiber cloth.

Once happy that you've buffed it the best you can.wash it again, and dry the car.
Repeat on any area's you think need another go until your happy with it or it's not getting any better.Wash and dry.

Now for the wax.
Use a new micro fiber cloth and keep it damp and the car out of direct hot sunshine.
Little circles and buff to a haze.Once done use a clean cloth and buff again to remove excess wax.

I'm waiting for the summer to do some areas of the current car.

A car I owned before took six hours but was cover in speckles of house paint and other unknown matter and once done the paint looked new and I had people asking me did I have the car touched up.