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I was wondering what the process is and what materials are needed to fill nail holes in newly installed baseboard and casing?
Depends on what material the trim is made of and what type of finish.
Painter's putty is normally used for this, be sure you get it for latex as the older types are oil-based and may bleed through. Alternately MH wood filler does well with this, but you need to keep the lid on the can or it will dry out quickly. A light press-fit is enough while you're using it, fully tight when you're done. Use a putty knife to apply that as it sets up fast and hard and can be a little extra work to sand down nicely and evenly if you leave excess. Latex can go right over it. Big-box stores have it as well as many paint stores. You'll get your best finish by using a latex primer under the finish coat(s).