Firenze (Florence) is a city in Italy.
Bus drivers are unconcerned by you having tickets, or not.
Parco di Rusciano (Southern side of river Arno, 20 min walk from city-centre) is a very nice place to pitch camp. Very quite area. Only one entrance to the park (or two, not sure). Very nice green grass to crash on, olive trees to set a hammock, salvage mint to make herb teas and laurel tree for cooking experiments. But it seems that Firenze is nicely surrounded by green areas, so you should be able to find other great spots somewhere else. Rainy-weather dry spots are a bit harder to find.
If you find yourself in the evening at the petrol station described above (direction: Pisa, Genoa, France) just go another 100m or so in the direction of a highway. You will find a 'green island' with one big and bushy tree. After the wall of leaves you will be positively surprised to see that someone has already prepared a lot of space to put up a tent ( exactly as I was on 16/08/2016 - whoever has done that - thanks a lot! saved me quite a bit of time!). A lot of cars make it pretty loud spot for a night though... --Polski Matematyk
If you are stuck at the highway entrance Firenze Impruneta (south towards Rome or Siena), you can walk back towards the centre of town following Via Senese/Via Cassia until you pass the big monastery La Certosa, on your left, on a hill. You will find a juction and, on the other side of the road from the monastery, you will find the squat La Riottosa. Just walk over the guardrail and ask if you can stay for the night, they also have space outside where you might pitch a tent.
The city has free wifi, check this map.
If you have European ID, you can register for free at the "Biblioteca Nazionale Centrale di Firenze" (and probably in other libraries) and you have unlimited time access to a computer with Internet. It's open from 8.15 to 19:00 from Monday to Friday and from 8.15 to 13.30 on Saturdays. Otherwise there are a lot a "Internet Points" in the centre, where it is eur 1 per hour (you can't ask for less than an hour), and they are open later at night and on Sundays.
You can also register at any municipal library, which is probably faster and easier than the National Library, including the Biblioteca delle Oblate which is 2 minutes from the Duomo (cathedral) and has beautiful view, sofas, and computers. Also has long opening hours.