Hi larabee, and welcome to cruise-chat.com!
I don't think it's crazy to cruise with a 2-year old. In fact, our son had been on three cruises by the time he was 2! Bob is correct, though, in that it will not be a romantic cruise. But, it can still make for a nice family vacation.
As far as the ship, there are very few areas that are off-limits to kids. We've brought our son to the dining room, show lounge, and other areas. When in the show lounge, we sat in the back in case we had to make a quick exit. For the dining room, we practiced longer meals at home and in restaurants in the months before we left. One important note (and this applies to any ship) is that s/he will not be able to use the swimming pools unless potty trained.
We have some great memories of those early family cruises. (Emphasis on "family.") My husband and I had a good time, as well. No cleaning, cooking for a week!
If you decide to go, I'll be happy to provide some tips.