I am by no means an expert - I've been skating a little less than three years. So take my advice as you will. But the important thing is to learn and talk to people about skating. These forums are an excellent place; everyone's friendly. Several people have their own skating blogs as well.
People generally start off taking group lessons and then eventually start private lessons when the skill sets demand one-on-one instruction. Group lessons are designed specifically to teach the basic skating skills that everyone needs to learn to progress in the sport. Your local rink will offer these classes. Learn to Skate USA has specifics on programs and what skills will be taught: https://www.learntoskateusa.com/
I'm not saying that it's not worth going straight to a private coach - I know people who did that for their kids. One-on-one instruction time is beneficial. But group lessons are a way of gaining the skills you need in a supportive environment at a fairly reasonable cost. Also, with group lessons you'll have the opportunity to work with a range of coaches. Choosing a coach isn't easy - you want to work with one who supports you and your goals and whose personality works with your own. Because a coach can be the most marvelous teacher ever, but if your personalities clash it will be of little help to you. Working with different ones will allow you to see what you need. Also, think about what you want to focus on - ice dance, singles, etc.? You won't want to choose a coach who knows little about your intended specialization. If you don't know right off, that's fine, too. It just might mean that you start lessons and decide from there.
Boots are important. You are a beginner - talk to a trusted skating technician at your rink, or a coach (whether you decide to go group or private immediately), or both. I bought boots when I started skating and ended up having to get new ones less than six months in because the beginner ones did not support me. Adults differ in boot requirements than children - we tend to be larger, for one. Something that will support a six year old will not necessarily support someone in her 20s. Size is also a consideration. Adult feet, on the upside, generally have stopped growing so you won't have to drop several hundred dollars every time your size changes. So poke around on these boards. Read blogs and reviews. Do your research. There are quite a few boot companies out there and the models do vary. You want to choose something that's best for your feet.
Something else to consider. Skating is a very expensive sport. Boots, blades (they come separate at higher levels), lesson fees, ice time, competition fees, testing fees, ... it goes on. You will want to develop a budget, especially if you intend on committing yourself to to the sport.
But above all - skating is a lot of fun. It takes hours and hours and hours of practice, but the hard work pays off every time you learn a new skill or have an amazing performance. Or just every time you step on the ice with a smile because you are doing what you love.
So a final summation - group lessons are an excellent place to start skating, hold off on buying boots until you've talked with someone trustworthy, and have fun.
(And don't feel bad about your age. Ever. I started skating at 28. A number of adult skaters start later than that. Age should never stop you from doing what you want.)