Don’t try to do it by hand. You will need a random dual action polisher taking a 6” pad. Much less chance of burning the gel coat than with a single action polisher if you aren’t experienced .
If the gel coat is quite oxidised then a use heavy cutting compound with a heavy cutting pad like wool. Then a lighter cutting compound and lighter cutting pad to finish. The heavy cutting compound alone will give a good finish especially on light gel coat.
The horizontal surfaces like the top of the cabin may need some sanding with wet and dry sand paper if the cutting compound won’t do the job. 800 wet and dry would be the largest grit I would use for very badly oxidised gel coat. I would try 1200 or 1500 first. Don’t jump straight from 800 to 1500 it’s very difficult to remove the sanding scratches from the 800 wet and dry. Be careful not to sand too heavily.
Wet and dry sanding is last resort and you really only want to do that once or twice at the most in the life of your boat.
The textured areas like the gunnels top of the hardtop and antI-skid can be done with a short bristled upholstery brush on the polisher. See below.
I used the Gyeon products but 3M have some good products as well.
Then you need to protect the gel coat with a wax or ceramic coat. It’s a fair amount of work but definitely worth it.