I used to live in Italy, including in Florence. Just a bit of an insight, but if you stand up at the counter at coffee bars, it's much cheaper since you don't pay the service charge for sitting down. Also if you want to go a bit off the beaten track, look for the YouTube videos by Mark Wiens. He did a few in Rome. Also there is an amazing ice cream place in Florence that is near Campo di marte train station called Gelateria Badiani. Amazingly good ice cream! If you avoid eating at the tourist areas, it is cheaper and you can usually find some very good and tasty lunch deals.
Perugia is a lovely place and not as touristy as Rome Florence or Assisi but still easily accessed by train from either Rome or Florence.
If you like cheese, local supermarkets have a fantastic selection. You could always buy cheese, meat, bread and wine and binge on a midnight feast! Also there are some really delicious chocolates called gianduiotti that are like Nutella bites. Not sure where the Eataly food theme park is but it sounds good! 
Be aware that there are many scammers about, and of course keep your bag near. I was robbed twice in Italy. Be careful at train stations or when boarding trains.
And check if you need to punch your train ticket on the yellow machine before you board your train. I was last in Italy in 2011, but you had to stamp your ticket before you got on the train or you would have to pay a fine. I believe there is a new train service from Rome to Florence that is a fast train, but I find that trenitalia fares are much cheaper than trains in the UK.