Here is what my setup currently looks like:
I decided to stick with my current Mountain Bike, a 2005 Gary Fisher Tassajara. I was originally looking for an aero bike but after talking it over with my wife and reading Your First Triathlon by Joe Friel, I decided to stick with my Gary Fisher. It doesn't help that my wife and I are big Dave Ramsey fans so logic and my budget prevailed.
In case you are new to all of this like me, the red tire on the back is a Zaffiro Pro Home Trainer tire. It turns out that the friction caused by your bike tire rubbing against the trainer can shred it (your tire) so whether you are using a road, mountain or other type of bike you should get a trainer tire.