If you take a private lesson, you could tell the coach you want to spend that lesson learning to falling gently, and getting back up. Or maybe a friend could show you.
Last I knew, that was officially the nominal first part of group learn-to-skate lessons. But some coaches don't teach it, or don't teach it to adults, because they think adults will be afraid.
I personally think they are the most important skills a person can learn before beginning almost any type of athletic activity, to avoid future injuries.
FALLING
Falling gently on the ice is relatively simple. (falling on Skis is harder, because the extra length of skis has more leverage to force you past the limits of your flexibility - e.g., you want both feet and legs on the same side of your body. But that isn't essential on skates.) You want to relax enough that your muscles aren't very tense (maybe the most important thing - I think a large fraction of injuries are created by your own muscle tension), but you still want to be able to control your trajectory enough that no hard bone makes initial contact - try to make sure soft tissue touches first.
And you want to roll or slide across the ice, rather than keeping the contact in one place, which would have to take all the force. I like to let the contact roll or glide to other parts of the body, avoiding my skull. E.g., for forwards and sideways falls, I barely touch my hands and/or forearms, and glide the contact onto the rest of the arms and body. For backwards falls, I roll backwards, straightening my legs as it progresses, to slow me down, so I don't hit my head. Ideally I would roll over a shoulder if going very fast, but I'm not fast enough for that to be needed.
It helps A LOT to be wearing clothing that touches all parts of your body that might scrape against the ice (e.g., long pants, long sleeve shirt or jacket, gloves), so you don't have any skin contact - abrasions and cuts happen easily, if you happen to hit a scratch on the ice. (If you get an abrasion, I suggest you wash it clean, and use a Band-Aid or equivalent.) It is easiest to start practice falling from a sitting or kneeling position, and work your way up to standing position, and later from jumps. You want to be able to safely fall in all possible directions (e.g., forward, back, left, right). I almost always wear gloves. I don't wear a helmet, but you could at first, if it makes you feel more comfortable.
The thing is, other people who might run into you while you are down are a hazard too. You want to practice this while there are very few people on the ice. Some times and places public sessions are almost empty at the beginning or end of a session.
If you do it right, there really is no reason why you should be hurt falling on the ice from a standing position, or a short distance above the ice. Human bodies have to be designed to fall gently, because we are two footed beasties. You probably did it as an infant, many, many times. A few infants do hurt themselves falling, but most manage. If done right, falling shouldn't hurt.
Lots of practice is essential. If you practice, you eventually develop reflexes that are fast enough to do everything right. OTOH, if you try to figure out what to do while it is happening, there really isn't time. At a rough estimate an adult has 0.7-0.8 seconds to reach the ice from a standing height. A well established simple reflex can occur in about 0.1 seconds, but you don't always want to fall if you get a bit out of balance - sometimes you can compensate a number of ways - e.g., by moving a leg and foot to stop the fall. Making that decision, if it too is practiced, adds about 0.2 more seconds. (There is some variation, person-to-person in these timings.) Still plenty of time to do it right.
OTOH, suppose you start to fall, and you haven't practiced it enough for the sequence to become a reflex. It may take a few seconds for you to recall the suggested sequence, or figure out which applies. By that time you are already down. And you may also panic and tighten all your muscles, which makes you much more vulnerable to injury. In fact, muscle tension can itself break bone. And you may land on hard bone - like knees, base of the spine, elbow, skull, and break it.
There are a number of variations on falls, which a coach who gets lots of practice might be able to show you. Though you might have a coach who doesn't believe in this. E.g., some coaches think you should never fall backwards. That would be an OK theory, except that it happens a lot in real life, so I think it makes sense to learn do it gently. I took my first falling lesson from a modern dance teacher. Other people use martial arts instructors - though combat rolls may involve more force, and require you to get up more quickly, so you can't relax as much. Some sports require you to fall on rough surfaces, where you need to avoid rolling across your spine, and some require you to throw yourself clear of a heavy object, like a horse or motorcycle. But ice can be gentle.
I wish I could do split falls. They are are truly beautiful, and you end up spreading the force all across the legs, but if you aren't flexible enough to split, keep your feet fairly close together on the ice - and also keep them on one side of the body on skis. (I once had a very bad muscle pull on a backwards moving ski fall, because the tails of the skis crossed, and forced my hips more open than my flexibility allowed. If my muscles were more flexible, that might have created an ACL injury instead.)
GETTING UP
On skates or skis getting up is simple, in theory. You roll onto your knees. (On skis, you put your knees on the skis, but skates are too short for that. But I still put them inline with each other. Also, you probably want to place your skis across the slope, so you don't slide as you are trying to get up.)
Then you put one or both hands ahead of you on the ice, or directly alongside and outside the knees.
You move one foot forwards.
Then I at least put the one or both hands on the forwards knee, and put some of my weight on it, to get a little extra force and balance. People with high flexibility or very long arms can keep it next to the forward skate.
You stand up, first on the forwards foot and leg, which drags the other leg up too. That requires balance, and you may not succeed the first time. So you may get extra practice falling. No problem.
Some people can get up with both feet in back of them, and/or only by putting their hands on the ice, without knee contact. Or they do a martial arts style roll. Those requires more strength, flexibility and balance. Keep it simple, at first.
