The station that we arrived in from Bologna was the Firenze SMN station. Firenze means Florence, and SMN is short for Santa Maria Novella, which means the new Church of Saint Mary. Trains in Italy rarely require you to book your tickets online early, but do buy at least 1-2 days in advance. However, Trenitalia, which is the railway company, offers promotions from time to time that can only be purchased online, so check whenever you can. To check prices of ticket and timings from city to city, visit here
Once we arrived, we stopped by the tourist information centre near the train station for some maps and had breakfast at a nice cafe while we waited for the 10am free walking tour. There are two each day, one in the morning from the West to East side as you see the places relevant to the Italian Renaissance, where Florence was heavily involved in. At 2pm the tour goes from North to South, following the influence of the Medici family, which used to be the most powerful family in Florence at one point. The walking tours are relatively shorter than other cities and take less than an hour usually. The starting point of the tours are at the Santa Maria Novella Church, near a small tower.
Notable places that you should visit include the Uffizi Gallery and the Boboli Gardens. These usually require an entrance fee of €8-10. I only visited the Boboli Gardens as I am not exactly into the arts. The gardens give a spectacular view of the countryside and of the city, which looks very similar to Budapest with the huge duomo in the centre of the city.