Paint and a roller works best on my opinion. Just do all your taping up first and paint the surrounds for.
Actually, taping up is a pia. You paint a stripe at the top of the wall, then use the roller to paint down to just above the baseboard, and then use the brush.. I work in 2 ft sections down the wall, and repeat the 2 ft sections to keep a wet edge.
Paint rollers and brush. Start with ceiling , walls then trim work. Latex for walls and ceiling, oil for woodwork.
Here are some fantastic sites with step by step instructions. It is a big job. Goes quicker if the house is empty. My house needs painting badly and I am dreading it as I have wall to wall and floor to ceiling furniture that is extremely heavy and not easy to move. Hopefully your situation is better.