What’s the size of your dog?
Law in Italy says that dogs are allowed in public places, they can get on buses and trains (but you have to pay the ticket for the dog too, unless the dog is in a pet carrier).
Under any circumstances the dog must always be on the leash, you must bring the muzzle with you (but you do not have to put it if the dog is cunning) and poop bags.
In spite of the law, many restaurants do not allow dogs to enter (small dogs generally are admitted).
In the big supermarkets, little dogs could enter and generally you can find this kind of trolleys:
(obviously dogs should never get in touch with food for any reason)
For the bigger dogs you can find an area outside the supermaket to leave him.
Dogs are generally not allowed in museums (but there are some exceptions).
There are many sites here in Italy for dogs owners, let me know which regions or cities you are planning to visit so I can send some links for you - and your dog Max of course.