Where can you buy a horse on The Elder Scrolls IV - Oblivion?

Answer 1

There isn't really anywhere you can 'put' them unfortunately. They will always follow you when you quick travel. And when you quick travel into a city your horse is put into the stable outside the city. If your worried the horse is going to die if you for example go somewhere dangerus you can get an invincible horse called 'shadowmere' by doing assassins guild quests

Answer 2

you can buy a horse in the stables located in front of the cities (except imperial city). All you have to do is go into the house right beside the stable and talk to the person inside or around about buying a horse. here is a list of the cities, what kind of horse available there, and how much they each cost:

Anvil-White horse 4,000

Bravil-Bay horse 1,000

Bruma-Paint horse 500

Cheydinhal-Black horse 5,000

Leyawiin-Paint horse 500

Skingrad-Bay horse 1,000

you could also steal a horse, but then your bounty will go up and it wont stay where you left it. it is worth the money in the long run.

you can get two free horses, one Paint horse from Prior Maborel in weynon priory. or inherit Shadowmere from Lucien Lachance after "the Purification" mission in the dark brotherhood