Most European "bucket list cities", especially Rome, are so overrun with tourism there isn't much real culture left to experience. Their inner cities are devoid of their original inhabitants thanks to short term renting and the treatment of their homes as pure capital.
If you want to get to experience real italy (or any country) you have to go into the smaller rural towns. Italians appreciate it a lot if you learn their language. In the tourist traps everyone speaks english anyways and will often prefer that over your attempts at italian/might not even speak italian.
If someone crashes into you, check your pockets and watch out in general. Lots of pickpocketers are woman/girls and well dressed, dont get caught off guard. If you rent a car, don't make it a pinyata. Leave the compartment open to show there is nothing to steal in there.
In Greece its not as bad, the rural areas are great and there are lots of sites spread all over the place. The Theater of Epidaurus is really great, you can still experience the aucustic perfection, a great place if you have someone with you that can sing.
In terms of manners, I can only say stuff based on what I experienced with american tourists so far:
-wear appropiate clothing, a tangtop, tennispants and flippers are not what you wear in puplic, especially fine dining. We refer to those people as visual pollution. (And if you are on the heavier side, please don't show us your roles, noone wants to see that)
-dont treat everything as an experience you paid for
-dont assume everywhere is like the US or expect people to know your customs, you are in their realm. Lots of american tourists appear very arrogant due to that.
-put your sunglases down, serving a table with not a single eyecontact is very upsetting and it comes off as very disrespectful.
-dont tell everyone youre american all the time, noone asked/cares (I have no idea why so many do this, do they want special treatment ?)
-if you are not interested in sth., don't do it. A lot of archeological or botanical sites outside of tourist traps have guides that do this due to their passion, they have feelings and are not there to entertain you.
In general just try to be reflective and observe how others do it and people react to you, most will not be offended at someone who doesn't know but tries to learn.
