What are the flight training options for a private pilot's license in the US?
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).