Absolutely: there seems to be an idea in the States that European cities are full of sophisticates sniggering at badly dressed Americans. Yeah, and we all dress like catwalk models - not.
Rule number one: wear what is comfortable for YOU, forget anyone else. It was very hot when we were there, so I wore knee length shorts most of the time, my wife culottes. Smart casual is the order of the day, who cares whether it is smart American or smart Mongolian ?
Rule number two, in churches be respectful. Would a middle aged woman wear beach shorts in a church in Ohio ? no, so she shouldn't do it in Rome.
There are dress rules for St Peter's and the Sistine but these seem to be ignored when we were there.
As regards footwear, sandals will be fine everywhere (but no socks for the gent !), but athletic shoes will also do. Restaurants want your money, not your dress sense. Obviously if you go upmarket then dress accordingly, but otherwise be yourselves.
Remember the story of Mrs Thatcher when she was prime mininister: at one of her weekly audiences with the Queen she was horrified to see that she was wearing a dress very similar to her Majness. The following week she told one of her flunkeys to call the Palace and ask what the queen would be wearing. Why ? asked the palace. On being told the reason the reply was 'Don't worry, Her Majesty never notices what other people are wearing'. Exactly.