What type of bike has the most similar feel to a stationary bike?
This really depends on the type of stationary bike you are talking about. There's the older style as well as bikes that resemble road bikes (at least in terms of body position), as well as some recumbent bikes. There are bicycles that match pretty well to all three of those stationary bikes.  However, it's worth mentioning that just because you are comfortable on a stationary bike, does not mean you will be comfortable on a bike with the same posture.  For instance the first bike I linked to is very much upright.  A bike like that would be fine for a quick trip to the corner store, but you would probably tire of it quickly if you were trying to get some serious exercise.