I've been taking lessons since I was 5. My mom is a music teacher, so I couldn't escape the clutches the music world.

a.) Is it imperative you have your own piano?
It helps immensely, but a small 88-key keyboard also suffices. So long as you can practice.

b.) What pieces they started you off on, and what kind of things they taught you in your first few lessons.
The basic fundamentals of music. Nothing technical or complicated, just the simple stuff you need to start out. There are a few exercises you'll have to do each day in the name of coordination and warm-up, and there will be some simple songs for you to practice reading music.

c.) How does a standard, new piano cost? How much did yours cost if you own one?
I have no idea. For me, it was free. You can pick up a decent keyboard however in the hundreds. I have a couple.