My sibs and I are starting to work on helping our parents through this issue; it’s likely they will need to move into this sort of place in the near future.
Independent living, AIUI, means you can handle eating, showering, going the bathroom, and managing your medications all on your own; the facility serves meals in a dining room, and may also take care of your laundry. They are of course also immediately on hand if you have some kind of emergency.
Assisted living comes in different levels/costs depending on your needs. They may bring you your medications on schedule, check on you several times a day, provide incontinence products, etc. Different facilities will have different offerings.
A skilled nursing facility offers cares for the highest-maintenance residents, i.e. people who need help getting dressed, using the toilet, being changed/cleaned if incontinent, eating, etc.
I’d recommend visiting several places in the area where you might like to live, and getting tours, brochures, and prices for the various services you’ll need right away, and the services you are likely to need in the future. The tours will give you a chance to meet some of the staff and also assess the state of the facility and its residents. Check out their events calendars to see what sorts of activities and field trips are offered to residents, visit the dining room to check out a menu and see what sorts of foods they prepare. Find out whether the basic cost includes things like heat, electricity, TV satellite/cable, and internet.
Popular places may have waiting lists to get in that can be months long with considerable uncertainty (since you’re waiting for current residents to move out or pass away), so you’ll want to ask about that too and consider it in your planning/scheduling.