Lugano to Siena

You can do this easily by train. It will take roughly 5-1/2 to 6-1/2 hours. I see tickets for early May on sale on both the Italian (Trenitalia) and Swiss (SBB) rail websites, but for the one departure I checked, the Italian price was much lower, at least for passengers without a Swiss rail card of some sort. If you do have such a card, that would presumably narrow the gap.

The Swiss show some additional departures, so if nothing you see on Trenitalia fits your schedule, you can look for other options on the SBB. Just be sure you've got "No Reductions" selected if you don't have a Swiss rail card.

There is a national timetable change on June 11 in Italy, the complete schedules of trenitalia trains after Milan will be gradually loaded on the system in the coming months. Use a date before for planning or wait the 12th of June.

High speed trains are always the first ones to be loaded, then come the long distance Intercity and last the slow Regionale trains (like those that go from Florence to Siena).
If the Swiss site is showing more trains than Trenitalia going from Milan to Florence and then Siena they're probably wrong.

Note that the Milan-Florence high speed line is served also by Italotreno.it/en , a private competitor whose discounted tickets have been cheaper than Trenitalia's for months.

You can go from Florence to Siena either by Sita (?) buses or by Regionale trains.

Many prefer buses because they stop on the top of the hill, while the train station is at the bottom and it takes 15 minutes via public escalators to get to the historic center.

Thank you! These were very informative and helpful!

You can easily travel via train from Lugano to Florence, with a train transfer in Milano Centrale, then you can travel to Siena via train or bus.

You can book your train tickets in advance at considerable savings on Trenitalia.com

You'll need to choose the train stations in Italian:

Florence = Firenze S.M. Novella
Rome = Roma Termini
Venice = Venezia S. Lucia