Brands aren't the issue because almost all brands use the same parts or the same parts spec at a given level.
In low-cost bikes you will generally find one-oiece (Ashtabula) cranks, or threaded bottom brackets (68 mm for road, 73 for MTB, I think 1:25 thread pitch for English ... don't mess with anything else (French and Italian are rare anyway.) Older bikes of a certain quality might have cotterred cranks. I think probably you can google these and see what they look like.
Almost all bikes in lower price ranges use comapitible parts everywhere else. What differs can be clamp size (different tube and handlebar diameter.) Also there are two generally used steerer tube/headset arrangements, threaded and Aheadset/threadless. You will need to look these up and learn to recognize ... but with older and lower-cost bikes, threaded 1-inch steerers with 22.5 quill stems will likely be the norm.
I used to pick up bikes from the roadside and slap all the parts together to make working bikes. I rarely had any compatibility issues. When you start getting more modern things get a little worse---basically with bottom bracket standards proliferating.