The price of groceries and drinking in bars tends not to vary drastically around the big cities in Europe, but the lower-than-average rent prices in Florence make the cost of living very reasonable! Public transport isn’t as comprehensive as some cities (particularly with the Germans, where Berlin and Munich have almost flawless systems) but it is fairly cheap - as long as you don’t convert all that saved cash into drinking money, your cost of living in Florence should be very manageable.
For a room in a shared apartment, you’re normally looking at under €500 (often closer to €300) which is less than half what you might pay for an equivalent room somewhere like Rome. If you choose to ride solo and get a studio apartment (or a regular one-bed place) then you’ll probably be pushing the €700-800 mark - for a student exchange, paying double the rent you need to is just not worth it so I’d look at sharing with at least one other person.
As a rule, students tend not to dine out in nice restaurants too often, so it shouldn’t be too disastrous to hear that dining out in Florence is pretty expensive. You’re probably looking at €25 or so for dinner with a drink - not extortionate, but this is for a "good" meal; you can go an awful lot higher!
Grabbing a quick bite to eat is another story. Coffee and gelato (mmmm!) are cheap, and toasted paninis will be around €5, though they’re quite filling. For grocery shopping, I’d budget for around €50-60 per week. You can live on less, but if you want to cook most of your meals and eat reasonably well, this is around the right level. Add your booze on top of that.
If you live anywhere near the centre, you’ll find that you can do most of your travelling on foot. The city isn’t massive, and because the weather is usually so nice, and the architecture is stunning, you’re much more inclined to walk than in other cities. Bus tickets are €1.20 each and last 90 minutes, so even the longest journeys around town will only cost €1.20 each way.
Both taxis and Uber are available in Florence. They’re both reasonably priced, but given the regular flow of buses and the walkable nature of the city, I’d only grab one if you were in a particular hurry to be elsewhere.