In Rome, small and medium-sized dogs can ride the metro, buses, and trams on a standard fare ticket — the same one a human passenger buys, with no separate pet fare. They need to be on a leash and wearing a muzzle, or contained in a carrier no larger than 50cm if the owner prefers that instead. On the metro and regional trains specifically, dogs are restricted to the first and last carriage only, and buses cap it at a maximum of two dogs travelling at once.
Milan's ATM network runs on broadly similar principles for pets on the metro, buses, and trams, though it's worth checking ATM's own published regulations directly if travelling with a larger dog, since exact enforcement and carriage restrictions can differ slightly from Rome's.
Guide dogs are the one clear exception across both cities: they travel free and are exempt from the muzzle requirement.