You should first determine what type of paint you are painting over, as it can make a difference in your results. Pour a small amount of rubbing alcohol on a cloth and rub it on the existing paint. If the paint easily comes off the wall it's an acrylic or latex paint. If it doesn't it's an oil-based paint.
You must use a primer before using a latex paint on existing oil-based paint because the new paint will chip off otherwise. First, sand and clean the surface. Then apply the primer. Let it dry for at least an hour, and then apply the latex paint over the primer. Latex paint also dries quickly so applying a second coat is not time-consuming.
You shouldn't encounter problems with chipping when using an oil-based paint to cover the old paint, but you still need to prepare the wall first. Clean the wall, then fix any nicks with putty and a sander. One coat of the oil-based paint may be enough if you are painting over a light color. If you plan to paint over a dark shade you may want to consider applying a primer first. Using a primer may allow you to fully cover the old paint with one coat, but you may still need multiple coats.
Prepare the room before starting to paint. Place a large drop cloth on the floor to catch any splatter, then tape over trim and moldings with painter's tape to avoid painting them. Use a brush to paint an approximately 3-inch band around the edges of the walls. This will allow you some breathing room with your rollers so that you do not accidentally paint adjacent areas. Cover the roller completely by rolling it in the tray, and tap it against the edge to let excess paint drip off.