I've found that foam rollers (in the hands of some) have terrible stippling/stipling? problems (bumps along wood trim, for example), so I hate them. This is either from a cheap roller, not enough paint on the roller, somebody pressing too hard on the roller, and not let paint adequately dry between coats.
When it comes to painting walls, you really do need to use a roller, but use a good quality fiber roller cover on a sturdy frame and sturdy hardcore detachable roller pole. This is why you actually do need a roller:
You CUT in the corners and along all wall seams as you go along--then before that cut-in paint around the borders dries you roll the wall or ceiling with your medium-nap roller using a W or M pattern to first get the paint on the wall, then rolling up and down to fill in the empty area. I work in 3 foot wide patches, but also put up more paint if the paint is thin, the walls higher than normal, or if I didn't load enough paint onto my roller.
I use a quality hair or synethetic bristle brush from purdy that will hold a moderate load of paint but I work quickly and have been a professional. For tighter areas/trim, an angled brush that is about 2 1/2" wide is best (also from Purdy). Don't buy disposable junk as you'll have loose bristles coming off and into the paint, as well as very little control or evenness.
Another note: Buy the best quality paint you can without being ridiculous. I prefer latex for everything but metals, and usually buy moderately priced paint (Glidden usually, sometimes Behr which I don't like as well) for interior and exterior painting. As a pro we used Miller paint for amazing results maintaining and painting two different college campuses full of classic buildings that are irreplacable (some as old as 1800 or earlier). For my home, I buy Glidden, and for primer sealer I buy either BIN or Zinsser Bullseye Primer-Sealer.
1) Prep the wall by wiping it down with a barely damp clean sponge or low-lint rag to remove dust, lint, debris etc. If you have mold or grease, you use a smallish bucket with a mix of degreaser called TSP (better to buy pre-mixed in a plastic bottle with green fluid inside, rather than using the powdered in a box), a small amount of liquid regular bleach, and hot water. About 1/3 of the bucket can be TSP, 2/3 hot water, and then what would be about 1/2 C of bleach for a small painter's bucket. Wipe areas with grease or mold down with clean T-shirt rags, let rest for 15 minutes, then wipe off with clean t-shirt rags that are damp with very hot water. Keep the room well ventilated. Allow walls to DRY absolutely.
2) Use a water or oil based spackle to fill in holes--fast-drying will be quick but crumbly, oil-based takes longer to dry but will be smoother. Once the spackle is dry but not "old" you need to get med-light grit sandpaper and smooth the spackled area so it is flat and doesn't show a disturbance in the wall (or make it look as regular and smooth as you can).
3) Take a clean T-shirt rag and dust the spackle dust off the wall and trim, whatever, below. If the wall is dry, the dust will fly off or wipe off easily--no need to wet down the wall. Some people prefer to use a barely damp cloth if they are in no hurry.
4) Any creased areas in corners that have more than a 1/4" gap you can try to use latex or silicone caulk to fill. MAKE SURE IT IS PAINTABLE CAULK. If the room is going to be subject to a lot of temperature fluctuations, you should use silicone (PAINTABLE). Next, you apply the caulk into the creases a little at a time in layers if it is fairly deep. Have clean damp T shirt rags or painter's cloths on hand to wipe off any drips or excess as you go along (don't procrastinate on wiping off drips or bumps. Once the area is filled in, you take a barely damp entirely clean T shirt rag and wiped the entire thing to smooth it off. Let wall and caulk dry completely.
5) Priming/Sealling: This is really essential when you are doing a new coat of any kind of paint. If you're using a dark paint, stores have a darker primer (usually in gray) that you will apply, but most of the time people will just by the stark white primer/sealer. Latex Primer for Latex Paint. I prefer to use the Zinnser Bullseye Primer Sealer: it is thick, has mildewcides, smooths out the wall, and dries beautifully. Lasts beautifully too. If I have something more complicated I will use Zinsser's B-I-N primer (we just call it BIN). This is your first coat of paint, not the color you wanted. You could consider definitely using disposable roller covers and disposable paintbrushes for BIN as it will pretty much ruin a regular paintbrush--dries too quickly and does not behave well during removal from your brush. Primer is very important in your bathrooms!
6) PAINTING: You'll be waiting for the primer to dry for a while then you will need enough coats of paint applied not-too-thin, not-too-thick, to cover the primer well with no thin streaks between the "cut-in" areas that were hand-painted and the areas that were rolled. Places that are too thin can be hand painted and blended in with your handbrush if they are along the line of the ceiling. IF they're big areas of uneven paint, you get out the roller and brush and blend or do a whole new coat.
7) Rolling popcorn ceilings: if you don't want popcorn falling off from the liquids in your paint, use BIN Primer first to seal it in. You apply it with a very thick nap fiber roller cover that you will have to part with eventually. That thick nap will help fill in creases and crevices without damaging the popcorn. Don't press hard at all. You may need to do more than one coat of paint.
8) WHAT KIND OF PAINT SHEEN DO YOU WANT?
This is one of the first things to consider but I'm getting to it last. If you have a bathroom or kitchen then you want a protective sheen, not some kind of dry flat paint. I recommend eggshell for most purposes throughout a house but if you have an older kitchen or bath with a lot of humidity and very little modern ventilation then you should go for as shiny as a satin to protect the walls from mold and condensation. You may have to wipe those walls down once in a while, but it's better than a flat-sheen paint that will let the plaster or drywall underneath from being damaged. As far as regular rooms in a house, most people want either the dry look (flat paint) or a bit of a sheen (eggshell).
High quality room paint will not be drippy. Please have it machine-mixed (shaken in a shaking machine basically) at the hardware store, even if it's old paint. If using old paint, but the can is intact, they will usually do it for free or a very small fee. If you have old paint, you can paint some onto a stirring stick until it resembles the wall of the room you are touching-up or recoating, then take it to the hardware store and they can match that older paint to make extra for you in the same sheen.
I doubt this answers all of your questions but maybe this will help. I don't believe in spray-painting rooms or exteriors. I've seen so many houses peeling on the outside with as little as 1 year since they were painted--all were not cleaned properly, prepped properly, sanded then sprayed down with water then let to dry, no primer/sealer.... People just start spraying uneven paint that clots everywhere and it peels off the dirt, debris, older paint, etc. Painting takes patience if you want to do it right. Keeping that in mind, if you ever do exterior painting you will need to know a lot more than I'm telling you here.
Be sure to buy quality drop cloths that are thick enough that paint won't go through, but not those disposable plastic ones that can cause terrible injuries (tripping, slipping ladders) and painting accidents (pulling away with the painter's tape off of a wall).
Look into getting the best painter's tape you can, at the right width as well as the rolls of brown paper that you can attach to that painter's tape on a dispenser that is a lot like the metal dispensers they use for regular packaging tape. If you have nice carpet or trim to protect, and don't want windowsills, door frames etc to get paint on them, that type of setup is best. Keep your painting area tidy so you don't injure yourself. Use roller frames in a 5 gallon bucket for roller work. Not those crappy paint trays (if you want nice even walls you stir up and roll your paint gently, not making air bubbles--not trying to make a fluffy batter but a nice smooth even layer of paint with no air bubbles in it). Don't whip your brush around in the bucket and make air bubbles (which can also help the paint look streaky too).
If you're doing wood trim: consider doing it in a higher sheen paint than what will be used on the walls. I always do. My 1924 home is loaded with wood trim throughout. All of the trim is painted with semi-gloss to protect it--or I have refinished it with many layers of polyurethane or varnish.
Just remember: paint is changeable but TRIM IS FOREVER. Not exactly forever, but most people do not want to strip or refinish thin pieces of wood trim (or thick). Choose something tasteful that will last for many many years and do it right the first time. Anything that is a real piece of damage should either be replaced or patched as best you can with a stainable wood putty that you put on smoothly and smooth/sand out around the uneven areas and edges before it has hardened. Once wood putty hardens you will have a hard enough time getting it off a spackle blade, let alone your wall or trim.
Most of my supplies I buy at Home Depot and there's not a thing in this list above that I haven't bought or seen there on sale recently.