Hey, thanks, ya'all!
OP, I started in my mid-40s as a self-rated 2.5. I'm 60 now. Looking back on it, there are a few things I would do differently, so here goes.
I started off taking group lessons and drill classes. I now think those situations are horrible for a beginner who only has so much time. By that I mean, you cannot go out and hit hundreds of balls because you'll probably hurt yourself, e.g. tennis elbow. The pros in those group classes are not there to help you learn good technique. And you won't. I would suggest weekly one-hour private lessons if you can swing it. If not, every two weeks. Instructional time is valuable to build muscle memory, so don't squander it in a drill class where someone feeds you three FHs, gives no meaningful feedback, and you go to the back of the line to hit three BHs.
OnTheLine gives great advice if cost is an issue. Doing a 90-minute semi-private lesson is a great way to improve. You get a ton of instruction, and you also get to hit balls coming off of your lesson partner's racket. Maybe introduce that once you've developed a relationship with a pro? Also, I am a big fan of "private clinics." This is where you get three others (preferably teammates) to do 90 minutes to two hours with a pro. This helps get you all on the same page regarding positioning and teamwork, which will win you a lot of matches.
Which leads me to my next point: Practice. Practice what your pro is teaching you. I went years only taking lessons and playing matches. It wasn't until I found a practice partner that I really made progress. How do you find a practice partner? Ask your pro to put you in touch with his or her other students. Right now, I am trying to straighten out my net game and FH, and I am using a practice approach I've never done before. I take a lesson and then go immediately to the indoor facility that has a ball machine. I set it to feed balls to work on the exact skill we just worked on (last week, it was stepping into low volleys on both sides). Lo and behold, I volleyed really well this week.
And how do you find a pro? The same way you find a car: test drive them. Because you're 60, there will be pros who will not take you seriously when you say you want to improve. That's OK. Keep looking until you find a good one. The pro is "good" if they focus on building one skill at a time with an emphasis on developing muscle memory. The pro is not good if they blow sunshine up your skirt and take your money.
I would also say it is important to do things properly from the beginning. Use the correct grip, always. Do not take shortcuts to get results now.
And finally, and perhaps most important . . . please do not become a thirsty rating-obsessed diva. It's not the rating that matters. It's the tennis skills. If you improve, you will move up, if that's what you want.
Good luck, and please stick around and tell us about your adventures!