If you want to buy with gold there is a prerequisite quest, but it's short and easy. It can be picked up in a bank in any vanilla zone, but has to be finished in the "home" zone for that character. It will reward a free Inn room at the finish, and open the list of purchase homes for you.

Many of the homes have other requirements though, where you have to have completed some achievement or quest in that town before you can buy.

Snugpod is a house in Grahtwood that you can buy without having to do a quest first. It's pretty cheap, and close to a lot of really useful things like a wayshrine and stables, guild traders, crafting tables and an Undaunted enclave.

Most of the houses can be bought with gold - although some are quite expensive. You can pick up a quest in your starting city, that will give you a small room at an inn to start, completely free. So I would suggest doing this first.

Via the Crown Store you can see all available houses (or you can also see them on the world map). When you click on them to preview you have an option to view purchasing options. Here you can find the different prices both in gold and Crowns.

But as said, start with getting the small room, and then look at which house you want to save gold up for

There are houses, But they are insanely priced in gold... I bought Moon mirth., That one seems reasonably priced.