I know about the house in Whiterun because you can get it right off the bat, but can you buy houses in other cities? Do certain quests or factions give players houses or the opportunity to buy them? Please, no spoilers if it involves questlines or plot elements.

I think if you become a Thane of one of the holds you get the option to buy a home in said hold.

Key word in that sentence was think.

If it's like Oblivion, it should be one per major city.

Yes, you can buy houses in other cities. But, most of the time you have to do something for the Jarl or his/her advisor before they allow you to purchase a home.