I love your enthusiasm! I hope you find the journey incredibly rewarding.
I’m with Scribbler. At the lower end of things, good riding is good riding. Look for a barn with lesson horses that suit your level and abilities. Talk to the trainer about your goals (eventually dressage). Talk to him/her about what you need to learn to work toward that goal. As Scribbler mentioned, you’ll definitely want to hear the trainer respond with something about working on independent hands and seat. Ask how the trainer would help you develop those skills.
Don’t be resentful or afraid to go backwards to move forward. Sure, once you can w/t/c on your own, it seems a step backwards to have your trainer lunge your horse while you ride. But you can learn a lot on the lunge, as it frees you up to work very specifically on one or two things at a time. Once you can jump, pole work can seem too easy (I know you’re interested in dressage; just giving another example). But riders and horses at every level (I’m talking up to GP jumpers) can benefit from pole work.
Ask potential trainers about their attitude toward you lessoning with others while you lesson with them. In my experience, good trainers encourage their students to ride with other trainers from time to time. We all teach and learn differently, and it is of great benefit to get different perspectives. It took lessons and clinics with a total of 5 different trainers to finally get me keyed in on fixing some long-standing problems with my position and seat. If a trainer says you can only take lessons with them, run away. They’re either money grubbers or they’re sure they have a unique handle on all of the correct answers (no one does!).
You say you can afford lessons. Once you get into lessons, or as you check into lesson barns, you may want to ask about half-leasing opportunities. Once-weekly lessons sound like a lot, but in my opinion, especially as a beginner, you need a lot more time in the saddle than that. Another ride or two a week working on things from your lessons is really valuable. Plus then you can do some fun and relaxing things that can also benefit you and the horse, like trail rides or playing around with barn friends riding bareback.
Best of luck to you in your exciting new journey!