As it stands currently, there are 2 types of housing in the game. There are ward houses, and there are apartments. Each server currently has a little over 5000 ward houses available. They come in different sizes(the size of the house is tied to the plot its built on), they have a dozen different themes and styles(with some degree of freedom to mix and match), and each exists as a single physical place(the inside is instanced, the outside always exists out in the world). Being a summoner, one would guess you started out in Limsa. You could visit the housing wards of Limsa(The Mist), and see each individual house(their yards anyway - not everyone leaves their door unlocked for strangers to visit). Leviathan being a north american server, i would hazard a guess(i havent personally toured to check) that 99+% of your ward houses are owned right now. This is true on basically all the north american servers. You can find european and japanese servers with plenty of open lots, but the american ones are almost all owned and when they do open up they dont usually stay that way for long.

Now apartments are another story. Even the most over-crowded server in the entire game has plenty of open apartments. There is an apartment building in every division and subdivision of every housing ward of every housing district. In addition to that, any free company that has managed to get a house can also generate apartments(free company private rooms are basically the same, but slightly cheaper). Now apartments are instanced interiors, with a fixed set of amenities outside for your use. You still have the ability to do a lot of indoor decorating, but the space is smaller then the bigger ward houses and there is no outdoor area for you to decorate.

So how do you get a house? Well, right now each member of your family could get their own apartment to decorate. You could all be in the same building(neighbors), or spread yourselves out all over the place(currently 4 cities have housing wards, which includes the apartments). Apartments are only available for personal housing, so if you want an FC house you need to get a ward unit. Sometimes you find people selling their houses, and i guess you can deal with them if you want. If the sale is for gil, its probly legal(though you will also probly pay more then the base-price for said house). If they ask for actual money, they are violating the ToS for the game(and you paying it would be too). Most of the time, these people are actually selling you their FC which owns a house(you then rename the FC once you kick them out and make it your own). Others are selling "relocation rights", which means they will give up their house on a given plot so that someone else can have it. You dont want that, because you dont have a house to relocate. When a house is given up(people dont want it anymore, relocation, transfer off of the server, get their account banned, or inactivity demolition) and the plot vacated, the game applies a hidden timer before the plot can be re-purchased. This is anywhere from 6-24 hours that the lot will look vacant, but cant be bought. If you already have a house, you can relocate your house to any vacant lot you find right away(and now your abandoned plot is the one with the hidden timer). Once that timer is done, the first person to try to buy the house(interact with the sign outside the empty lot) gets it. Usually when a house opens up, youll see a handful(more on busier servers) of people sitting around the sign, clicking on it regularly and hoping they end up being the lucky one. I dont advise this behavior, but it has gotten many people their houses.

Instead, i woud say you should look to the future. They have added more housing wards several times in the past. The last few times they did, the wards were only open for FC houses and transfers(private or FC) for a few weeks before they became open to the general public. So you already having an FC, you could be using this time to save up your money and decide what area you would prefer a house in. Then the next time a new ward is added, one of you(with FC house purchasing rights) log in early on the day the housing goes live and try to get a plot you like. On slower servers, youll still find open lots for days afterwards(some lasting all the way until the public can buy them). On a busier server, the longer you wait the less choice youll end up with. Its also worth mentioning that theres an event going on in game right now(playing out over a couple patches) where we are rebuilding part of a war-ravaged city. The area we're rebuilding looks a LOT like a housing district(which would make it the 5th city to have one), and theres a lot of speculation that when this is all over new houses will open up. Nothing confirmed yet, and this might even lead to some sort of change in how housing works. Now none of that means you cant keep an eye out for a house now(and like i said, you can each have apartments in addition to houses), and good for you if you get one. But rather then obsess over it, id try to plan for the future.

So i just typed a lot. If you have any specific questions feel free to ask. Someone here is bound to know most of the answers. And welcome to the game!