True story...
...last year on Black Friday (the day after Thanksgiving), I was wearing my flat American Eagle sneakers. Going out on the road starting at 5:31 AM CST to about 10:30 PM CST was a long excursion, even while heading home to safely drop things off at home. By walking around malls and shopping centers, I've learned something- flat-soled shoes are only comfortable for so long. At one point, instead of taking steps, I was basically sliding and scuffing my feet on the ground while walking. I was able to comfortably walk again after a few days and while wearing shoes with actual cushioning.
So to recommend flat-bottomed shoes or something (flat flip-flop/thong sandals, ballet flats, or any flat shoe without sufficient cushioning) is almost a no-no for comfort. I may consider getting some more comfortable sneakers or something. I actually have a pair of Skechers that I've stopped wearing after getting a lot of sticky stuff on the sole back in January. So I think it's paramount to get some shoes with sufficient cushioning and support. That's even if you have to wear some Dr. Scholls with those or something (assuming they aren't sandals or some kind of open-foot shoes). As for STYLISH shoes for traveling, I don't think I can answer that, at least speaking as a homme.