This is written from the perspective of a retired professional pilot.
There are many options, but the only one I have had any experience with is American Flyers at Morristown, NJ. Our daughter completed the majority of her training for a Private Pilot Certificate in our Piper Cub. That airplane is not equipped with an electrical system, navigation radios, gyro instruments, or lights, so she required additional training in an airplane thus equipped in order to attain the unrestricted certificate.
American Flyers accurately assessed her previous training and then provided everything she needed and nothing she didn't to complete her training and attain the Private Pilot Certificate. Her flight instructors did not have a great deal of experience but they provided good quality instruction and she was well prepared for her practical test. We found the company to be professional and efficient.
Whatever training organization you choose, you are about to invest a breathtaking amount of money in the enterprise. It is not necessary and may not seem relevant, but getting a tailwheel aircraft endorsement somewhere in your training will make you a more proficient pilot whether you are flying a Cessna 172 or a Boeing 747. Andover Flight and Tailwheel Academy in Andover, NJ can provide that, although I have no experience with them. The cost of that training is relatively small and provides big benefits. If you can get your initial training in a tailwheel airplane, all the better.
Another consideration is aerobatic training. Again, this is not required for your objective, but well worth the investment. A five hour training course is sufficient to make you a better and more proficient pilot. Without that training, any airplane you are flying is more capable than the pilot flying it.