Unpainted wood and drywall can pose huge adherence problems for coatings without primer. With fresh drywall, the substrate actually absorbs a coating really well, like a sponge. However, because it absorbs so well, without a primer coat, you'll experience uneven coverage, especially along drywall seams.

Once the right primer has been purchased, feel free to prime your drywall. Lastly, after 2-3 hours, you can safely paint your drywall.

The general rule of thumb is to properly tape and mud drywall joints before painting. Not only will skipping this step lead to unattractive walls, but it is also against building codes. So yes, mudding the drywall before applying the first coat of paint is a necessary step.

Drywall and skim coats are highly absorbent and will require multiple layers of paint. It is better to apply one to two coats of a quality primer and seal the pores in those substrates before your top coat.