There are two main Puccini museums, coming from the division of hereditary assets. One is in Lucca, and is the home where he was born. Paraphernalia exposed here mainly from the legacy of the Puccini family (he was the fifth maestro in line) and a treasury of interesting letters (you need to know Italian and read Puccini hand, that is more difficult than knowing Italian). He owned the home all his life, but due to an affair with a married Lucchese woman he could not live there.
The other museum, and I would say the main one, is in Torre del Lago near Viareggio, being the home he built himself and where he lived from 1900 to 1919. Puccini's son Antonio had the idea of burying his father there, being the place most dear to his heart. A small chapel was built out of a junk room and the maestro is buried in the left wall, that is the place closest to the piano where Butterfly, Fanciulla, Rondine and Trittico were written. (I had the honour of playing the piano for a museum extraordinary opening and knowing Puccini is right there on the other side of the keyboard made the pressure unbelieveable.)
From 1919 he lived in a big villa in Viareggio, now much dilapidated and cannot be visited; Turandot was written there (but the Turandot piano was moved to Lucca). - Getting to Torre del Lago (by bus from the railway station in Viareggio) may be time consuming but if you are serious about Puccini IMHO you cannot miss it.
The Verdi villa in S. Agata is exceptionally beautiful and interesting, but it will be closed till an hereditary lawsuit is closed; now it is bound to be auctioneed and it will be a long and expensive thing, may not open again unless the state impounds it. - There is a smaller Verdi home in town, actually the home of his patron Barezzi, and the very small home in Roncole where the maestro was born, in front of the church where he played organ. - Visiting these places with a car is easy but not too much with public transportation as they are several kms. from each other. - I am not too keen on the museo nazionale at Villa Pallavicino that was opened only in 2009 so it is not as authentic as the other sights.
In Venezia, I would pay hommage to the Monteverdi tomb in S. Maria dei Frari; in Florence I would visit the Rossini tomb in S. Croce. The small Teatro della Pergola in Florence is mostly like it was in 19th century when it hosted Rossini, Donizetti and Verdi premieres. The main venue for opera in Florence is the Maggio Musicale Fiorentino, in a modern building. In Venice, Teatro della Fenice was painstakingly rebuilt after the fire some years ago. It hosted several Verdi premieres and he liked it more than La Scala (that he did not like at all for most of his life) and the Parma opera house.
