Now, each monster is composed of different elements. Some are cold, water, plant, rock or some combination of elements. This becomes important when you start breeding them to make new monsters. Combining monsters of a simple makeup will yield the best results (a rock with water, or plant with frost, etc.). If you breed two that have multiple elements you may just end up with a duplicate monster which isn't the worst thing in the world, but you have wasted a bit of time. (Of course, that's the point.)
You do have the option to speed things up by spending diamonds. You only start with 15 diamonds, and you can burn through them surprisingly quickly. And it's hard to make them up. You can buy them with actual real-world money - not the coins that your monsters are making for you.
In the meantime, you can build bakeries, upgrade your castle, clear away rocks and trees, buy decorations that your monsters like (seriously, if you get them something nice they make more coins for you) and play various games. Eventually, you'll be able to even buy new islands for monsters. The game itself is free and available for iPads and most smartphones. If it ever becomes available for PC, I'd be on it like white on rice. I do have a hard time playing it on my phone as it is a bit of a battery suck.