☆.A.
Before you try to paint the vanity, you need to determine what is on there now. If it's wood, you're going to have to remove any finish, old polish, varnish, etc. And rough it up with sandpaper. Tell the guys in the paint dept. what is on there, take pix with you & they can tell you what type of paint would be best. For example, if there is oil-based paint on there now, you cannot use latex pain on top--it will bubble.
As for the sink--you're going to paint the actual sink? Smallish new sinks and faucets aren't that expensive, so I would probably replace the sink and hardware for a nice update!