Yes your dark trim can be painted. Do you know the type of paint or finish on your trim? It is important to identify the type of paint on the trim as water based paints can not be applied over oil base paints without a primer first. When in doubt it is best to use a primer.
Painting with a light color, like white, over a dark color will take a few coats. To lesson the amount of coats needed you can use a flat paint then top coat this with your preferred sheen.