Hi @porra60,
If you need to colour-match the paint, then you'll have to scrape off a paint chip so you can take it in-store with you for colour matching. Something the size of a 50-cent piece is usually what we recommend. Do not worry that you are causing further damage because you will be fixing the damage anyway. Removing another paint chip is a necessary evil.
You're going to want to start by applying a very thin coat of a filler such as this UNi-PRO Multi Purpose Interior Filler to fill the holes in the paint. If you don't apply a filler, then you will be able to see a difference in height between the layers of paint. Once dry, give the filler a light sand with a sanding sponge and brush away any dust from the frame.
After this, apply a primer, followed by your topcoat of paint. Door frames typically use enamel paint like this Dulux Aquanamel as they tend to be high-wear surfaces, and it is a harder paint than what you would use on walls.
You can apply your paint and primer using a paintbrush or roller, depending on your preference.
Check out How to paint trims for some further guidance on the process.
Let me know if you have any further questions.
Jacob
