You don't automatically get expansions placed. You have to do it yourself. Click on the four arrows at the bottom of your screen, next to the $ symbol. You'll get a screen like this:
The roped off areas (5x5) are the directions you can expand to. It is recommended that you keep your city in as close to a square formation as you can. As you mouse over them they'll highlight in green around the edges.
Once you pick your desired location for expansion, click on the highlight area. You'll get this screen:
Pay close attention to the areas the red arrow is pointing to. If you don't have enough coins, it will be grayed out and you can't expand. If you hover over the thing, the info box will pop up, telling you how many expansions of that type you have. There are three options for expansions: Premium (buy with diamonds, and the price goes up every time), research tree (buy with coins, limited amount per chapter), and province fights (capped at 400-something provinces, it's an odd number).
Just because, here's the same screenshot with the province information: