WA-HOO!! Come to the dark side, honey!! It's a slippery slope, taking lessons. Next thing you know you'll be back here asking for opinions on the horse you're planning to buy. Hee hee hee...
For now. You should decide whether you want to learn Western or English (of some kind) because not all lesson barns teach both. It doesn't matter so much what kind of English you want to do, if it's English, because the basic skills are the same when you start no matter what discipline you go for.
What to look for in a lesson barn:
Clean facilities without a lot of dangerous looking metal stuff and other junk lying around. There will be SOME junk but it shouldn't be the kind to make you worry about getting cut up on it.
Horses that look healthy and happy (note: all horses get pissy, even the happy ones). Horses shouldn't have prominent bones and ribs you can count, and should have reasonably bright eyes and even, shiny coats. You can't tell anything useful from manes and tails because a lot of that just comes from the breed, and some horses are really hard on their manes and tales. So just because those look ratty doesn't mean the horse is unwell.
A lack of barn personnel beating horses with sticks or using whips to hit them. Using the end of the lead rope is fine, and using whips in the air is part of lunging and certain types of training, so just because you see a whip come out doesn't mean there's abuse going on.
Clean-ish tack. OK to be dusty. Not OK to be frayed or encrusted with filth.
Ring in good condition. Ask yourself if you'd want to run on that surface. If there's a big ditch where you can see that the horses have been walking in circles around the edge of the arena, that's not so great. That happens, but it's not great for the horses' legs. It should be getting flattened out regularly with a harrow.
What to look for in an instructor: someone with experience teaching people like you (some people do great with kids, some with adults, some with both).
Someone who will take time to answer your questions.
Someone who will teach you basic horse care (grooming the horse, tacking up, all that stuff is part of "riding").