Use 12" or 14" knife. About 1-1.5" of mud and apply in an X pattern. Buy the mud in a bucket that you need to add water to. Consistency should be smooth, not watery. Practice makes perfect.
If you want a smooth surface look: just tape and blend seems, sand, then prime & paint. Depends what look you want. Hope this helps!

Prime first. You should be good to go.

Only skim coat if trying to match the rest if the house. Sounds like if the whole bathroom has new drywall, then no need to skim coat.

A full skim coat or "putty" coat is a Level 5 finish for smooth wall. Sounds like you currently have Level 4, where some paper shows. We use a PVA primer on drywall (polyvinyl acetate). Start with that and then two coats of your finish paint!

Thank you all very much.