Ukemi is the basic foundation of Aikido, Judo, Jujutsu, etc. IMO, it is one of the most practical aspects of Aikido because you will more often trip and fall than be faced with a self defense situation. Thus, learning how to take falls without injury great self defense in itself.
So, rolling/breakfalls/backrolls are vitally important to developing into a good martial artist. Ask your instructor if you can practice ukemi more than you usually do, and when you start feel comfortable enough to do good ukemi, you should be able to practice it at home (in the house probably isn't a good idea). The grass in my backyard seems to be fairly easy on my body. I just have to watch out for dog poo. (Adds a new element to training).
I sometimes do ukemi on concrete or other hard surfaces to practice...I wouldn't recommend this, though. Although, the first few rolls I did I could tell something was wrong because it hurt, haha!
The best thing I can tell you is don't be in a hurry to reach the mat; you can exercise control in a fall/roll just like you can in a throw or joint lock.