Trenitalia generally releases schedules 2 months out, so for June they should be coming soon. Check around April 19 or 22, considering those are around your travel dates.
You can still use schedules to check what is available in mid June - check how long the train rides are, the places where you might have to transfer.
KM-wise, it might be less distance to take train to Chiusi and then either train or bus to Siena.... if you can find a train that works for you. Time-wise, there might be more trains available to Florence, which might make total travel time shorter. It really depends! You might also consider taking bus the whole way - there is Flixbus that has affordable bus rides. It might actually take less time to take Flixbus straight, rather than going around to the north to Florence and waiting for train connections to head back down to Siena.
As far as getting to Lake Como, yes, you will need to take either bus or train to Florence, take train to Milan and from there a regional train to Lake Como.
As far as tickets go, you can download the Trenitalia app and buy your tickets on the app. It will remind you to check in and "validate" directly on the app so it makes train travel easier. All other websites buy tickets from Trenitalia and resell them, so it is the same trains.
The fast trains are called "frecciarossa" or "frecciargento" while all the slower trains are either Intercity (IC) or regionale (regional, slower trains but that's because they make lots of stops to serve more towns along the lines).
You can always buy paper tickets - look for the yellow machines near the train tracks and stick the ticket in them, they print out date and time on them to show you are validating them for use. This is more important for the regional tickets, since tickets are sold generic without a date or time on them, they can be bought today and used on any train whenever you want, within two months of issuance. That's where the date and time stamp on them is important, to show you've used it for a specific trip (and can't be reused every day). For the fast trains, you get sold a specific date and time train, with assigned seat so validating them has remained a necessity of travel although they can't be used on another train. The validation shows you used the ticket, so can't claim that you missed the train and need to catch the next one. If you do miss your train, or know you won't be using it, depending on the ticket you can ask for a train change BEFORE the time of travel. Depends on the type of ticket you buy. You will see conditions of the fare you buy once you're ready to buy tickets. You can browse and pretend to buy a ticket on the trenitalia website and see what you find.
Hope this info helps! Let me know if I can help with anything else.