I am italian but I have been abroad for quite a long time now. I try to give you some advice, but better double check.
As you pointed out, there are 4 major carriers: Vodafone, TIM, Wind and 3.
-TIM has a ~100% coverage but it is bloody expensive
-Vodafone used to be the best, but they increased the price plan for no reason. As far as I know, they have just launched a 12 euros/month plan (on top up, no contract) for unlimited SMS and 1 GB full speed internet, but you still have to pay for the calls
-3 has the best plan out there, starting from as little as 4 euros. Network coverage is kinda poor and if you are roaming on other networks you have to pay extra (and the extra is quite a lot). If you are in a big city, then this would be the way to go
-Wind used to be crap but now it is becoming the most appreciated network in Italy. Not the cheapest, but very fast internet and impressive network coverage. They now have a very intersiting offer, 300 minutes, 300 SMS and unlimited internet for 10 euros a month. I would strongly recommend this one.
Hope it helps