The historical center in Florence is really small and Villa Antea is not far outside of the "viali" which mark the outer perimeter of the center. So you're definitely within walking distance from all of the main sights. As you head from Villa Antea towards the center, you'll first reach Piazza San Marco - the southeastern corner of the square has the Academy of Fine Arts and right next to it is the
Accademia Gallery where you can visit and see the original copy of Michelangelo's David (you'll also find a copy in Piazza della Signoria and another in Piazzale Michelangelo but definitely the original is the best).
From there head south toward the
Duomo (Santa Maria del Fiore). Here depending on time and whether you feel up to the challenge, you can either climb all the way to the top of the dome (about 440 steps, no elevator!) or to the top of the bell tower, which is right to its side.
From here then head further south along Via Calzaiuoli and you'll arrive at
Piazza della Signoria, where you can see the
Palazzo Vecchio, the Loggia dei Lanzi and then pass by the
Uffizi Gallery and arrive to the Arno river from which you can have an excellent view of
Ponte Vecchio.
You can do all this on your own - unless you want to have historical explanations, dates and such as you see each place, these are all walking distance and doable within the day. From the Arno, you can either walk up to
Piazzale Michelangelo or take a bus from along the Arno. Here is a map that shows the various walking routes and buses:
Getting to Piazzale Michelangelo on foot or by bus
As you walk downtown, you'll find ample opportunities for shopping - and there are many good restaurants where you can have authentic Tuscan cuisine as long as you stay away from those in the main center. Ask at your B&B, I am sure they will be able to recommend some really good ones both near the B&B as well as further away that won't be too "touristy". I am sure you'll have a great time in Florence no matter if it is just a few short days. For train schedules, take a look at
www.trenitalia.com so you can plan your travel times from Rome and to Naples.