In July, we will be using Florence as our travel hub for about 2 weeks. Our plan is to travel only by train and foot. I would love your suggestions on places to day train trips to/fro Florence.

Lots

by train- Lucca, Pisa, Bologna (vastly underrated city IMO with what is claimed to be Italy's greatest collection of historic monuments and buildings in its old city center); the Cinque Terre; classic hill towns like Cortona, Assisi and Perugia and Siena (Siena better IMO by bus - cheaper and quicker and takes you right to the top of the hill - not below it as trains do) and Montecatini-Terme, one of Italy's most spiffy and famous spa towns - right by Lucca could combine with it.

I think you should consider traveling by bus as well, since many good day trips are best done by bus rather than train.

What are your interests? What do you hope to do and see on these day trips? Knowing this would help us make better suggestions. For example, I recently took a day trip from Florence to Poggio a Caiano to see the Medici Villa and Still Life Museum there. If I had been interested seeing in a quaint hill town, it would have been the wrong choice.

I like Orvieto by train, Siena and San Gimignano by bus. The bus station in Florence is just across form the train station.

I prefer Lucca by bus as only old R trains serve Lucca and the buses are nicer and have AC. You can also reach Volterra by bus.
http://www.sitabus.it/sita-toscana/Firenze-Siena08.pdf

I would also check if the Florence Tourist Board is still sponsoring the Thursday tour to Chianti for 15€ .

Any particular reason why you won't use buses? Siena (and other towns) is more easily reached by bus.

Not a day trip and not accessible by train but Fiesole, up above Florence, is a wonderful town with great views of Florence as you return. But it's a long uphill walk if you won't take the bus.

Siena and Pisa are both reached more quickly and easily by bus from Florence.

Although I've not actually been to either, I did do quite a bit of research a couple of years ago, and both Volterra and San Gimigiano are easily accessible by bus, less so by train.

A nice - and short - bus trip from Florence is up to Fiesole. Absolutely worth it for the views.