I’m presuming you are American. The UK has a very different horse culture from the USA.
UK ‘stables’ or ‘yard’ = USA ‘barn’.
Livery generally comes in 4 types: ‘full’ in which all care is provided by the staff, including exercise, ‘part’ in which the owner does an agreed amount of the care and staff do the rest, ‘DIY’ when the owner does everything and ‘working’ when the horse works off a proportion of the livery costs by e.g. being used in school lessons.
No one leases horses from a trainer, but loans and share arrangements are often made between riders and horse owners which range from a friendly agreement through to a formal contract with agreed financial contributions. Horses simply do not have the same financial value in the UK. You can easily buy an experienced eventer or hunter for less than £10K. You can pick up a good honest cob for less than £1K.
In the UK, only qualified vets can legally treat animals, and all other therapists such as chiropractors can only work with the approval of the vet. There are many specialist equine vets and referal practices for more complex problems. Similarly, only qualified registered farriers can shoe a horse. Two consequences of these two monopolies: drugged horses at shows are vanishingly rare and UK farriers consistently win the world championships. On the other hand, any Tom, Dick or Harry can treat humans!
The BHS approves livery yards as well as riding schools and these are listed on the BHS webpages by geographical location. This should be your starting point as approved yards will all meet a minimum standard and then you can look in person to see about the extras and unique features on offer. Be aware that the animals will probably be a bit shaggier and probably show, umm, more character in their ground manners. We Brits think horses are fun and expect the animals to do all kinds of interesting things with their humans - an all rounder is good. Eventer’s compete in dressage, dressage horses go hunting…
If you intend to bring your own horse, it will depend on the particular animal how many and what weight rugs will be needed. Aberdeen is on the coast and that will moderate the any cold and the east of Scotland is far drier than the west.
There is lovely riding country to hack out in but Aberdeen is quite far north for any major equestrian competitions. Scottish riders travel much further to show than do their English sisters.