I moved out at 16, and I went to a hostel (it's sometimes called supportive living). the YMCA are the most widespread hostels, and operate in a lot of different areas.
You usually need to do some research, to find out what hostels are near you, and gather some contact info for each of them. Sometimes hostel waiting lists are quite long, so by contacting a few different places, you decrease the time you'll be waiting.
The next step will be to ring up the hostels, and arrange an interview with them. This is always a good idea, as you can get a feel for the place, and see if you'd like to live there.
Rent wise, housing benefit (or universal credit) covers all of it or a percentage of the fee if you're working. As it is supported living, you will be assigned a keyworker, who has meetings with you to make sure you're doing okay, and you can bring up any issues you are facing. Your keyworker will also help you apply for housing benefit.
With some hostels, you have to share certain living spaces (like a bathroom & kitchen) but you will have your own room. Other places have en suites in their rooms, and have a small kitchenette for each individual resident.
I would definitely have a look into it, as once you have lived there long enough, they will help you apply for a flat.
Hope this helps!