Here’s a calculator to help you determine how much paint you’ll need (if you’re making a drastic color change, or covering a darker color with a lighter color, you may need two coats).
You should buy a 9” roller frame and a couple of roller covers (the fuzzy part), one for each different paint you use. 3/8” nap will do for most interior surfaces. Buy a broom handle (or steal the handle from a broom at home) to screw into the end of the roller so you can paint the ceiling without a ladder.
You’ll need a brush for “cutting in”, which means painting the corner areas where the roller can’t reach. A 3”-wide brush is about right (get bristles labeled for “all-purpose” use). You’ll go around and paint the edges of the ceiling or wall that you’re working on (as well as around widow and door frames) with this brush before filling in the rest with the roller. Also buy a metal roller pan and several plastic liners for it.
Get a couple of rolls of 2” wide painter’s masking tape (I like the green stuff, but the blue works well too). Use this wherever you need a precise transition from one kind of paint to another (once you get good at cutting in with the brush you can skip the tape, but that’s not easy).
Get yourself some disposable vinyl gloves (you’d be surprised how hard it is to get paint off your hands), a box of painter’s rags (unless you have a bunch at home already), and a cheap plastic drop cloth.
For most wall and ceiling jobs, don’t mess with washing before painting. Unless it’s super dirty, just paint right over it. When you switch between paints or take a break, wipe the excess paint from the roller cover (using your gloved hand, then toss the glove) and wrap the roller cover tightly in aluminum foil. This will keep it usable for a few hours. any longer than that, you’ll have to wash it (which is a major pain). The foil trick will also work with your brush; if it’s idle for more than ten minutes or so, wrap a damp paper towel around the bristles, then foil.
I think that should do it. Good Luck!