Pompeii may be a stretch, unfortunately, but I've always wanted to see it. That and the Archaeology Museum in Naples with that amazing Alexander the Great mosaic. Sirmione looks to be a bit of a pain to get to. Pisa doesn't excite any of us much but it feels like we'd be amiss going all the way there and not seeing the tower...
Sound reasonable? Any must-dos? Anything obvious missing? I'd love to see a real Italian castle, like the one in Sirmione (Castello Scaligero). We're definitely doing the Vatican, ancient sites in Rome, museums in Florence (and getting one of those 72h passes), walking between the towns in Cinque Terre, and doing the touristy stuff in Venice (including Gondola ride).
There are some Etruscan ruins outside of Rome that I have heard called the Pompeii of Rome and then you might get your castel fix in Rome too at the castel sant'angelo. They might be cheap fixes since both Pompeii and Sirmione are well worth seeing, but both seem to take up a bit of time that could be spent better. Depends on the trains though, if you find some nice high speeds then go for your plan.
Edit: also I'm not a local, so let them chime in too!