Go to a good school. They will match you to a quiet horse. Wear a very good helmet and get it fitted. It's safer than a borrowed one. If jumping, wear a body protector. Wear boots that aren't too wide so they can't get wedged in a stirrup and with decent ankle support. Tall boots give better grip. Sticky riding tights will make rising trot much much easier.
My advice as another adult beginner is not to go in expecting to never fall. You WILL fall at least a few times when learning and probably will come to no harm whatsoever but there is of course a possibility with any sport that you may be hurt.
They will spook. They are epic wusses. They spook if a leaf looks at them funny. But it's usually over quickly when they realise the tinsel they walk past a million times a day actually won't eat them. You won't always fall and they won't put you on a bolter.
I've stacked it twice. Both times off the quietest most reliable horse in the school. Once because another horse knocked a pole down behind him and he jumped and spun (he was a bloody showjumper ffs! Don't tell me he's never heard a pole fall before!) And once when he spooked at the canter because a deer ran across the field.
They spook differently too. The one I fell off tends to drop his shoulder and shies sideways so he just sort of disappears from under you. I got a lot better at staying on him by riding him bareback. Phenomenal for giving you a good seat. But his spooks can be hard to sit. He is over it quickly and you invariably find him stood next to you looking a bit embarrassed.
The one I ride most now just sort of flinches, gets his bum under him and scoots which still makes your heart go but he does at least take you with him. If you do fall, he generally stops and waits. Most school horses will. They don't generally teach beginners on horses that are prone to panicking.
If they do spook, tighten your reins up, sit up, push your heels down and relax your bum. They can feel your every move and will follow your lead. If you clench up and go tense, they start looking for the reason. You relax, they relax. They won't go far, likely just flinch andbits over. It feels bigger to you than it actually is.
Resist the urge to lean forward. You are most stable when sitting tall with shoulders back, leaning forward throws off your centre of gravity.
Give reassurance and praise. Regular pats, good boy etc. Sing quietly if you're nervous or talk softly to them. I tell them about my most unreasonable customers.
Ears pinned back flat means back off. Stop what you are doing or I will bite/kick you, I don't like it.
Pick a school who teach horsemanship as well. So tacking up and untacking your own horse and grooming them yourself. The more you handle them on the ground the more you will grow in confidence and the more they will respect you.
Where I go, you can hang out and help before and after lessons so I usually catch them, groom them and tack up before my lesson then if I'm last, put their rug on and tuck them up for the night. It's good for bonding with them. My first horse there follows me around like a dog. He loves attention and runs to greet his riders who spend time with him.