Instructor should ask what you want out of the lesson and actually listen to what you say. They should also pick a good horse for you (i.e., a quiet one that is used to carrying green riders). Some may disagree, but IMO, a riding lesson should include grooming and tacking up, not just riding. That grooming/tacking process is important for bonding with the horse and getting everyone's head (yours and the horse's) into the right space for riding. If it's the sort of barn where you'll groom, then the instructor or someone employed by the barn should catch the horse for you and put it into the crossties for you, and then introduce you to the components of the grooming kit, show you how to use them, and then let you do it. They might just tack up for you at first, since that's a safety issue, but even if they do tack up for you, they ought to be teaching you how to do it yourself.
The horses ought to be in good physical condition (no ribs showing, looking generally clean and happy), and cooperating with the handlers.
You'll probably spend a bunch of time just walking around on the horse. This is important, and not at all a problem. Even expert riders work their horses a lot at a walk. Just like with a ski instructor, the riding instructor should be helping you advance but not pushing you into places you're not comfortable. When you're skiing, the snow doesn't care how you feel about what you're doing, it's just the snow...but when you're riding, the horse definitely cares about how you feel about what you're doing, and will change its behavior as a consequence. And don't think you can hide what you're feeling from the horse. Riders have NO secrets from their horses when they're astride.
The horse will not necessarily change its behavior to be more accommodating to you, either, it may change its behavior by getting naughty. If this happens, your instructor should realize at once what is happening and tell you how to make it stop. That doesn't mean you'll be able to, any more than you can take a bump field fluidly on demand when you're learning, but the instructor should be giving you relatively clear instructions.
If the instructor is yelling it doesn't necessarily mean they're "yelling" at you - most of them seem to yell habitually to be heard over the racket that horse and tack can make.
Tell the instructor your goal is to eventually join the adult lessons, and ask him/her to let you know when you're ready. You may need to discuss this a couple of times, depending on how flaky your instructor is (and the odds are high that s/he will, in fact, be kind of flaky), but they ought to let you know when you're ready. There is no standardized amount of time for how long this will take. Some people will be ready to do it almost right away, some will take a lot longer.
Riding once a week is a minimum, IMO, for keeping your skills moving along and your muscles developing. There are muscles you use when riding that you don't seem to use for anything else, so you will be sore after the lesson (even if all you do is walk), and I don't think there's much you can do in the gym to keep that from happening. Best way to condition yourself for riding is to ride.
You're going to need some gear, of course. As soon as you decide you want a second lesson, you're going to need some boots. If you're riding Western, it's cowboy boots, if you're riding anything else, it's paddock boots and half-chaps. They're $60-$100 or so, and available at any tack shop or online. You'll probably also want some breeches, regardless of what discipline you pursue. If you're on the smaller side, you can get these at the tack shop. If you're generously proportioned, you'll probably have to get them online. You also, of course, need a helmet, also available at the tack shop. The barn should have some loaner helmets for brand-new riders like yourself, but as soon as you decide you want to go more than once, buy your own.
That reminds me...the instructor should not permit you to get on the horse without wearing a helmet. Period. No, you cannot use your ski or bike helmet for this. I checked. Ski helmets are designed for relatively low-speed collisions, and both them and the bike helmets are designed for falls from no more than six feet. If you fall off the horse (and this does happen, but it is not something to be particularly worried about), you'll be falling from quite a bit higher, and the riding helmets take this into consideration. Horses are at the bottom of the food chain, and species survival is contingent upon reacting very quickly and strongly to perceived threats. Some of this can be trained out of them, but not all of it, not any more than people can be trained not to blink if something heads right at their eyes. It's reflexive. So even experts, riding horses that they know extremely well, still wear helmets.
This is going to be great, by the way! I remember my first riding lesson like it was yesterday. I couldn't wait to go back. I haven't been able to ride my horse in almost a year because I injured my piriformis, and it's taking FOREVER to heal up completely, and it's driving me nuts not to be able to ride him. More nuts now that ski season is starting to fade off here.