Mugello lies to the north of Florence so it is a great place in which to base yourself, as well as explore, during your time in Tuscany.
We really recommend renting a car if you're going to use the Mugello as your base, since it will make it easier to explore the small towns and villages in the valley as well as those spread out into the Appenines, such as Marradi and Firenzuola.
It is possible to get into the Mugello valley through various roads from Florence. The quickest way into the area is to take the Motorway A1 (take the Barberino di Mugello exit) while the most panoramic road into the area is through both the Faentina (SS 302) and Bolognese (SS 65) regional roads from Florence.
Once you're in the valley, many roads connect the small towns making it the best way to move around.
You can easily reach Vaglia, San Piero a Sieve, Borgo San Lorenzo, Viccchio and Dicomano by train, as the tracks make a wide loop from Florence in the direction of Borgo San Lorenzo. If you stay in these towns, then you can get by without a car to get back into Florence and then from there move on into different directions.
From Borgo San Lorenzo, a line also heads up to Marradi before heading to Faenza. This is mountains and trains go every 2 hours so not very often. If you want to explore the smaller towns and areas of northern Mugello not along the line, it won't be easy.
Buses travel from Florence to Borgo San Lorenzo. There are buses to smaller towns nearby, such as Scarperia, but service is limited, particularly on the weekends. So while you can get into the area by bus, it won't be easy to move around using the bus system.