I recently visited Venice. I was there for less than three days but managed to experience quite a bit of it and get a feel for the city. Here are my notes from that stay. I've also included a map with the locations mentioned.
While roaming charges tend to be high (€2 or $3 per MB), there are prepaid SIM cards in most countries. One of these will usually set you back a mere €10 and offers from 1 GB and up of data transfer. This is usually enough for at least a few days of use. But instead of putting it in your phone and using its hotspot feature, and not being able to keep your home phone, buy a portable WiFi router. These come with built-in battery which lasts up to five hours and provide a local WLAN network which up to five devices can connect to.