Ballet flats and what I used to call "Nixon shoes" (those little Mary Janes, usually made in China) are great for house wear, but people with flat feet in particular need to be careful about wearing them for much walking or being on your feet for a long period of time.
Oh, definitely. They're "hotel slippers" -- or "out to dinner slippers," at most -- exclusively.
For extensive walking you need a substantial, sturdy but flexible, athletic-shoe-type sole and a top that doesn't pinch or rub. Since the days of the little old ladies in Miami Beach, though, they've developed shoes that offer those comfort features yet are still relatively attractive -- or at least don't call attention to themselves as unattractive. These are all Privo brand (photos from Zappo's).
Keens are another brand that are pretty reliably comfortable. I got a pair for my first trip to Italy and wore them for years after that. Mine were lace-up, but here is a pair
http://www.zappos.com/keen-golden-ballerina-potting-soil?zlfid=111&recoName=zap_brand_page that appear to be decent for walking but are also reasonably attractive, or at least inobtrusive, depending on your taste -- I know there are some women who only find delicate high-heeled shoes attractive, but I'm not one of them.
When in Italy, I constantly saw women at tourist sites -- including places with long lines, extensive walking, rough terrain -- picking their way around in delicate, high-heeled shoes or boots. They didn't look fashionable to me; they just looked pathetic. It's also pathetic to bring so many shoes (or clothes in general) that your bag is really heavy as you're lugging it through train stations, a mistake a couple of my friends made on the last trip. My formula has been two pairs of good walking shoes, one lace-up and hike-worthy, and the other also sturdy, flat and comfy but slightly dressier, like the Keens above. Plus the hotel slippers.
But even lace-up, sturdy, comfy shoes can be tricky. Years ago I brought a pair of lace-up walking shoes to Prague. I'd only worn them once or twice before that. My feet got so blistered I either had to pass on certain things I wanted to do, or do them but in total agony. The moral: wear the shoes around for at least a few weeks before your trip, have a shoe-repair place stretch them out if any parts are tight.