is the old paint cracked, chipped or lifting? sometimes you can leave it on if its in decent condition
sometimes paint strippers can be handy, use a rag and spread it on bits at a time, let it soak in a bit, and then scrape it off with a putty knife. quick sanding after to finish up anything left behind
sanding works well too, but be careful if its old paint, it can have lead in it
You can get some paints with primer in it already. saves a step. you also need to know if the paint is oil based or water based for your choice in strippers, primers and paints
small roller brushes can be nice for larger surfaces like doors, if there's not a lot of detail, lets you make a more even spreading/coverage and removes some of the paint brush strokes that can show. there was a nice one i used recently that mentioned on the label it doesn't show brush strokes, made it real easy to lay on and keep smooth. might have been a behr indoor paint made for trim and doors