Decide what you want to do with the baseboard. If the floor was worn out, chances are the vinyl base is old too and it is not difficult to remove it and put fresh base up after the floor is done. If the baseboards are wood, you may decide to repaint them, or protect them with blue painters tape applied very carefully.
Clean the floors completely. A heavy duty cleaner like TSP will be your best bet. If the floor has ever been waxed, buy some wax stripper from a hardware store. If the floor isn't one hundred percent free of old wax, no paint will ever adhere properly.
Sand your floor. A palm sander will work very well, but if you don't have one the sanding sponges in coarse grade will work fine. You are sanding to dull the floor and provide "tooth" for the primer. Don't be tempted to use one of those "liquid sanders", they will not work well at all. Wipe down the floor before applying primer.
Use a bonding primer like KILZ or XIM. There is no need to do more than one coat of primer, ever. Brush around the edges of the flooring, then roll a good coat of primer over the whole floor. Let dry according to manufacturer's instructions.
Choose a floor paint wisely. You will have many choices, from water based paint, to two part epoxy coatings to oil based deck paint or even marine paint. Good floor paint will not be cheap and it is worth the money to get a coating that will hold up to heavy wear, moisture and repeated washing. Your best course of action is to go to a real paint store where real painters go, and talk to the knowlegable staff. Do not even think about getting decent floor paint from a "big box" home store. They don't have any and the staff are generally not well informed.
Follow the manufacturer's directions for whatever coating you buy. You will have to do two coats of whatever it is. Always brush first and use a mohair roller for a smooth paint job without any little fuzzies and lint from the roller. Cheap roller covers are a false savings because they simply will not lay down a nice smooth finish.