From Milan, high-speed trains (Frecciarossa, Frecciargento, Frecciabianca, Italo) allow for day trips to many cities in Northern Italy. You can get to Venice in two and a half hours, to Padua and Vicenza in two hours, to Genoa and Verona in an hour and a half, and to Turin, Parma, or Bologna in one hour. And if you purchase your ticket well in advance you may take advantage of economy and super economy fares.
If booking online, be sure to choose as your point of departure and destination two stations where the Freccia makes a stop: if you have to change trains and take a local or regional train as well, the online system may not offer the economy and super-economy fares. Just purchase the tickets for the local and regional trains separately, either online or at a train station.
Local and regional trains are slower and cheap, and they can take you to beautiful places like Bergamo, Cremona, Lake Como, Mantova, Stresa (Lake Maggiore). Regional and local train tickets must be validated immediately before you board your train, by putting them into the small green or yellow machines at the entrance to every platform. There’s a heavy fine if you don’t—up to €200. You do not need to validate tickets that you purchased online as they indicate the train you will have to board (within the next four hours you may also board any other regional train to the same destination). Do not validate Frecce tickets (and some InterCity tickets) as these are only valid on a specific date and train. In case the stamping machine does not work, go to the ticket office or inform the ticket collector while boarding the train. Please note that tickets must be validated shortly before your departure as from the moment of validation they have a validity of a few hours.
Please do not board a train without a ticket. Purchasing the ticket on the train would require a surcharge of €50 or more.
To plan a trip and to purchase tickets online, go the Trenitalia website. [Please note that the official and legitimate site of Trenitalia is trentitalia.com, NOT thetrainline.com]. There is no need for reservations to board local and regional trains and no discounted fares are available.
You can also find high speed trains on the Italo website. Most Italo trains leaving Milan depart from the Milano Porta Garibaldi station and not Milano Centrale.

From Siena, it is possible to take a regional train to Florence or to several small towns in the area, but the bus lines are much more extensive and cheaper. Some buses leave from the train station, which is a bit of a walk from the city center, but many buses, including the ones that run to Florence or San Gimignano, depart from Piazza Gramsci, which is a short walk from Piazza del Campo, right in the center of town. See the Tiemme website for schedules. Buses to Florence are faster than trains, provided you take the “rapida” instead of “ordinario.” Click here for the schedule of buses from Siena to Florence and return. [< This link will become active as soon as the summer 2023 bus schedule is published]

Another cheap way to travel in Italy (and Europe) is by bus, with low cost companies like FlixBus.