Are you planning on rolling right off the plane and onto the train? Personally that seems like a lot on top of an international flight and I would recommend thinking about staying one night in Rome before heading to Venice.
Either way Rome airport is an hour outside the city and then Venice is another 4 hours away after that. Fast trains can be purchased at a discount ahead of time but the discounts come at the cost of flexibility. If you're going directly to Venice then I would wait and purchase the next train as you arrive in Rome Termini. If you're going the next day you can buy the earliest ticket that you feel like committing to at a discount.
Venice is amazing and aside from dock>boat transfers for the local "buses" on water the city itself is flat aide from bridges.
Venice to CT is a solid 8+ hours from door to door as you are crossing the country coast to coast at almost it's widest point. You might consider Venice > Florence and then onto CT afterwards if an entire day of travel doesn't thrill you.
Pisa and Siena are both easily reachable via train and while Pisa is flat Siena is a hilltown and requires a certain amount of up and down as you go around the city. Train from Florence to Pisa (and maybe consider Lucca as well) and then train or bus from Florence to Siena. The bus drops you off more centrally than the train which is on the flat but there are escalators and buses up into town.
Assisi is hard to get to and IMHO too far for a day trip from Rome. If you really want to see Assisi I would detour to Perugia between Florence and Rome and stay there and visit Assisi from there before proceeding onto Rome. Note that this will mean taking the slow trains for this segment rather than the fast trains the run only between big cities like Florence and Rome.
You'll want to use the official site of Trenitalia which is the state train company of Italy: https://www.trenitalia.com/en.html
They also have an app for use in country that also let's you track trains in realtime and validate tickets etc. Use the Italian city names for best results when searching on train sites: Roma for Rome, Venezia Firenze for Florence and get used to 24 hour times for train use.
.Italo is a private train company that runs only fast trains between big cities. https://www.italotreno.com/en
Local train tickets - called regionale - do not come with seat reservations, never sell out and always the same price so buy them when you need them. Tickets in the app need to be purchased a minimum of 5 minutes before the departure time and validated in the app a maximum of 1 minute before the departure time. Paper tickets are available from train machines and these should be validated in the stamping machines as you approach the tracks.
Hope that helps, have a great trip!
=Tod