Can you suggest a city in Tuscany that is convenient for taking day trips to Florence, Venice, Perugia or Assisi? Also, we will not have a car so what mode of transport would be best for this travel?
This is a tough request as nothing is really central to what you want to do and many Tuscan towns do not have rail stations close to the center of the town. Should you choose a hill town like Montepulciano, there is added time to get to the station outside of town by bus as it is not walkable.
Take a look at www.Rome2Rio.com and plug in various combos, i.e., Arezzo to __________ and see what you can live with. We try to keep the one way public trans time to about an hour. We find if the roundtrip train journey for a day trip is 4 hours, it is just too much and we'd rather stay in/closer to the destination.
How many nights will you have?
In some ways I think Florence works as a base city. It's a major train hub for regional and Freccia lines. Perugia and Assisi will be about 2 hours (Arezzo and Lake Trasimeno are along the way) as will Venice (but on the Freccia line with possibilities of Bologna and Padua). From Florence you also have easier access to Lucca, Pisa, Verona and Siena (I recommend Siena by bus). Go to trenitalia.com and check out schedules and prices to see what you are comfortable with.
How much time will you have? Those locations do not lend themselves easily to one central base from which you have good public transport options to all of them. If you have a week, I would suggest two bases. Bologna works well for good connections to both Venice and Florence. Then transfer to Perugia for a couple of days to visit there and Assisi.
Sometimes it helps to look at maps.
Florence is between Perugia and Venice, almost half way, therefore it is the most logical base.
However the distances you are trying to cover are not short.
Florence to Perugia is the same distance as Oklahoma City to Tulsa.
Florence to Venice is the same distance as Oklahoma City to Dallas.
Thanks to high speed trains you can go from Florence to Venice is a little over 2 hours.
There are no high speed trains between Florence and Perugia, but the distance is shorter, therefore the regular speed train can get you there in less than 2 hours.
Venice is a pretty expensive day trip. Trenitalia does have a discount "same day return ticket", but it is a discount off the full fare.
http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=0e81aae8de63c410VgnVCM1000008916f90aRCRD&vgnextchannel=fe013041aeb2c410VgnVCM1000008916f90aRCRD