I think I have settled on the New Balance Minimus 20V-3 Trainer as my all-purpose exercise/work-out shoe. Lightweight, stable for weights, good for everything a Cross-Trainer or gym rat might need. They do not make you feel bare-footed, but close enough.
They have a 20v4 out now, but the 3 is fine. They come in men's and women's because men and women are different.
New Balance (made in USA) tend to run 1/4-1/2 sizes small, so I go up 1/2. They make narrow, which is good.