The most efficient way to paint and room: if you are painting  just walls, or walls and trim or the ceiling, walls and trim is the following.

Skip what does not apply.

1. Ceiling first. Complete two coats and smush the roller into the corner with the wall being careful not to leave three-dimensional drips on the wall. When the ceiling is dry, if you bought quality paint and it was the same color as your paint, only one coat is needed in most cases.
2. Then you paint the trim. Use a 9 inch roller and bucket or pan. Look on this website for what a professional setup looks like. It’s a 5-gallon bucket with a screen hanging from the rim. Using a roller pan is OK but it’s slow and messy.  Dip your roller you roll out a section of the trim and you brush or what we call tip out the paint. You do that even right against the wall not worrying about getting trim paint on the wall but no heavy drips
3. When your trim is finished one or two coats whatever you need, then you switch to wall paint.
4. Just so you know switching from white ceiling paint to white trim paint which is usually semi-gloss can be done without cleaning the roller and the brush. You just squeeze them out very well and then put them in the new paint and squeeze them out again. You only really need to clean when you change colors.
5. Painting, the walls might seem unusual, but this is how we do it: first role all of the surfaces careful not to hit the ceiling and the trim. Generally, we put masking tape on top of the trim, but since it’s got fresh paint, you have to be careful that the tape will not pull off the paint later. Masking tape is generally meant for only a few hours and not many days. The green tape is generally waterproof and has glue that will not pull off paint but you have to be careful. I put my masking tape up on the wall a little bit and then brush just above the baseboard. I also lay tape above the door frame. After you roll the wall twice then cut twice (or once with a good brush and good paint). This all depends on the color change if any. If you’re dramatically changing color, you need a primer first. You could do three coats of your finish paint but that finish paint is very expensive compared to the primer so we often use a basic drywall primer.  With no stains on the wall and no special cases, because you are just changing the color use budget primer.
6. After the walls are rolled, and you see the coverage is suitable then you start brushing the corners. Cut right up to the trim and the ceiling. If you have the paint for the ceiling and the trim, you don’t have to be enormously careful because you can always touch up mistakes. That lets you go faster. A quality brush will cut a fast clean sharp line. A cheap brush requires tape and doesn’t work anyway. It’s never worth buying ‘budget’ tools. Bad brushes shed bristles and are hard to control and frustrate you. Same with rollers.