Formalities to bring your pet in Monaco

You need a passport for your pet with indicated all the due info - microchip number and general vaccines including rabies.  Monaco does not require rabies in itself but it is mandatory if your pet travels. No quaratine as far as I am aware if you come from an EU country.  If you are flying in contact the authorities at Nice airport and they will be able to advise. Once in Monaco you can go to any Vet and have your dog registered accordingly with your new address. If you want they can also register it with the central French system so that if your dog gets lost anywhere in France they can easily track you down.  All does not take very long and it is not very expensive.

Dogs are generally welcome in Monaco and France and Italy.  Great walks near by.  Beach areas dedicated to people with dogs (not in Monaco but in the town next door). Unfortunately there are no fenced areas specifically dedicated to dogs so if your dog has a tendency to run off then you may have a slight problem to find a spot where he can run and play. 
Dog walkers/Dog sitter available if needed. 
Good Luck