I'll take a swing at this, mostly because I started out a long time back with some of the same questions. I read some good reloading/handloading books, took some experienced advice, and then started trying it.
how strict must you be with regard to brass and primer brand? You don't need to be because your rifle; chamber and barrel; isn't matching theirs, anyway.
Of course my Nosler manual is going to call for Nosler brass, how big a deal is it to use Federal or Remington brass? It's not, as long as you follow the good advice given above and start lower in the scale and work up.
Same with primers... if one manual calls for a CCI 200, can I safely sub in a Federal 210? Depends on the load. If the load is near max, you'd need to back down and work back up to make sure you're safe.
Or must I adhere to the manual to a tee? No, see above.
You can minimize the number of components you purchase. Pick your brass. Pick a primer you can readily get from your shop of choice, online, whatever. Pick the powders you want to try. Pick the bullets you want to try. Start low or low mid-range in the book data and work with those components until you find a load that works and be happy.
Having said that, if you change those chosen components, then like some of the other guys said, you have to back down and work back up. Different brass can make a BIG difference in pressure and a max load can be dangerous if you switch to a case that generates more pressure than the one you worked it up in to begin with. Federal and Remington are an easy example. Using a .308 example, Federal is heavier, with less volume, and generates more velocity in every one of my .308 loads. For a specific example, 48.0 gn is at or near max of IMR 4320 and it is right up in the neck of a Federal case, but at the base of the neck in a Remington case. The difference is visible. Both are safe in my rifle, but the results on paper and velocity are different.
Speaking for my own experience and that of the couple experienced hand loaders I associate with, brand will usually matter with POI and group size results, as well. Sometimes, you might get lucky with similar POI and only a small change in group size, other times it might be drastic, but there will usually be some kind of difference. Primer changes things, too, sometimes, though not as often from my own testing. So, it will matter in that respect. I typically try a few different brass manufacturers once I settle on a good, safe load and choose the one that gives the best accuracy for that load. It's not always the same brand, or even the most expensive brand. Unnecessary, if you're looking to minimize, it's just an enjoyable excuse to shoot more.
Excellent and thoughtful response! Thanks for taking the time!
