Do I have my steps right for painting over stained trim?
Because "it barely had any glossy look to it" and you sanded it and cleaned it well means that it has been properly prepared! Good work.

Feel free to prime and paint it now, but do yourself a couple favors and buy good quality paint and professional grade brushes. I cannot stress how much a difference using quality applicators will make. Just build time into your schedule for thoroughly cleaning them and they will serve you well for years.