Italy then Portugal?

I'd gamble on nicer temperatures in May than in June for the southern Italian part of your itinerary.

Why pick countries so far apart?

If you pick Italy, then why not visit Switzerland, Croatia or visit the South of France instead of flying to Portugal/?

If you pick Portugal, spend five weeks in Portugal and Spain.

While in Portugal, don't miss Porto in the north as well as the amazing Douro River Valley.
We did a 7 day river cruise there and it was fantastic. If you don't want to do a cruise, take the train up the river to Regua.

Paestum is really far from Sorrento. You'll have three full days--one for Pompeii, one for Capri, and one for the Amalfi coast. You could see Paestum from Naples. I don't know the travel time, but it's much easier than from Sorrento.
I'd probably put the lakes time into the Siena stay, but otherwise looks pretty nice!

No time in Florence?

If you're using public transportation, I'd just point out that it takes a while to get from Rome to Siena and from Siena to Venice. Staying in Florence (saves you about 2 hours train time from Rome and again to Venice) and daytripping for a horseback tour (is there such a thing?) would be more efficient, although I agree Siena is a nice town for a couple nights.

For the southern Italy portion, I'd split the time between Naples and Salerno which is actually on the AC. Visit Pompeii from Naples by the Circumvesuviani train and maybe ferry to Capri (it's never been a draw for me, especially as a day trip - maybe go to Ischia for a day from Naples instead?). Then take the fast train to Salerno (and then to Rome - much easier than schlepping to Sorrento) to see the AC and Paestum.

I'd go straight to Sintra from the airport and spend 1-2 nights to have a full day there. Uber is not expensive and it's much faster and easier than public transportation. Transferring from the metro to the train in Lisbon is complicated. Going to Lisbon afterward by train is easier, though uber probably won't cost much more and will take you right to your hotel.

I love both Portugal and Italy, but to me Italy's sights are a little more impressive. Ignoring weather, I would be inclined to have done Portugal first as I think it may seem a bit anti-climatic after Italy.