3 weeks in Italy

That's a very good itinerary. You can see the high points in that amount of time in my opinion, other than Sicily (which really requires at least 4 days minimum).

With 3 1/2 weeks, here's where I would go and about how much time I'd allocate to each:

Rome -- 3 to 4 days
Naples/Pompei -- 1 day (enroute to Sorrento for the night; stop long enough in Naples to at least see the Archaeological Museum)
Sorrento -- 2 to 3 days including one day on Capri
Venice -- 2 to 3 days
Florence -- 2 days
Siena -- 2 to 3 days, including a half day in San Gimignano
Pisa/Lucca -- 2 days (stop in Pisa for a few hours enroute to Lucca and spend 2 nights and one day in Lucca; be sure to walk or bike the wall encircling the city)
Cinque Terre -- 2 to 3 days (hike between all the villages!)
Lakes District/Lake Como -- 2 to 3 days
Milan -- 1 day

That adds up to 25 days if you take the "high end" estimate I've provided. If you want to slow it down some and spend more time in any of these places, the first places I'd cut from the list above would be Milan (although you'll probably fly out of there and spend some time; if not, you could just stop and go to the Duomo and Galleria enroute to the Lakes), Lucca and Lake Como in that order. If you don't have to go to Milan to fly in/out, you could just skip Milan & Lake Como altogether. That would free up around 6 days. After that, I'd hate to have to choose to give up any of the remaining places and would just move along quickly enough to work them all in.

Just consider Sorrento as a base for daytrips to Pompeii (and I would add to the island of Capri as you are there).You might also head further 100km southwards and reach the archeological area of Paestum (with Greek ruins) that is well connected via railway and also trying to sunbath there (on the west side of the town there are free and private beaches, but are usually overcrowded).
Otherwise for a seaside experience you might reach Viareggio when you are in Tuscany (the private beaches are about 1 km away from the railway station, the free ones can be reached via a local bus). Also I would add Siena during your stay in Tuscany.

I wouldn't bother with Milan or Pisa but would add in Sicily.

So assuming 23 days:

5 - Rome
5 - Sorrento/Pompeii/Capri/Amalfi
5 - Florence/Sienna/Tuscany
3 - Venice
5 - Sicily

I agree with #3 about Milan/Pisa, but I would rather spending more time in Tuscany, especially the area around Siena and then Lucca. Give a look to the following options:

Southeast of Siena:
day-trip) Pienza, Montepulciano

North of Siena:
day-trip, but could be more) Chanti area (and it’s wine!)
day-trip) Monteriggioni, Colle Val d’Elsa (high area of the town), San Gimignano

South of Siena:
day-trip) Montalcino, Sant'Antimo, Monteuliveto, Murlo

From Siena to Lucca make a stop in Volterra