In the USA, what training options does a student pilot aiming for their private pilot license have?
There are lots of options. Flight training is the US is usually divided into part 61 and part 141. Part 61 is general flight instruction and any FAA approved instructor can teach you under it: many part 61 instructors are independent or work for or with FBOs at GA airports. Part 141 governs flight schools that have a standard curriculum and are much more structured; they're often associated with training commercial pilots looking for airline careers. The license you get is exactly the same whichever way you learn.
There are pros and cons to both approaches, this article summarizes them well but the most important general point is probably that part 61 is 'one on one' training so it can be tailored for you and fit your schedule, whereas part 141 is 'school' training and if you can't follow their schedule then it usually isn't possible anyway. For that reason, part 61 is often the only practical option for many private pilots who have jobs and families.
Perhaps your real question here is "how do I find an instructor?", in which case you need to search the web, check out local FBOs and flight schools, ask for recommendations from pilots you know and so on. It's definitely worth investing some time to find the right instructor for you, because that can make a huge difference to your progress (and your costs).
Practically, the biggest variance in cost is going to be the price per hour of the airplane you rent.
A brand new Garmin 1000 equipped 172 might be 175 bucks an hour plus. There may or may not be a fuel surcharge on top of that. A 1968 Piper Warrior with basic instruments might be 100 an hour fuel included.
For basic VFR training you don't need a shiny new airplane with lots of bells and whistles, so if you're shopping for price, shop based on the price and availability of rental planes. Instructor fees and other costs won't vary much.