Estella,
You shouldn't have any issues with poor network coverage while roaming in Italy or elsewhere (referring to your post in the "France" section), as you'll be using local carriers while there. Depending on the specifics of the roaming arrangement negotiated between Sprint and other carriers, your phone will usually seek the strongest signal. That means you may be roaming on TIM in the morning and one of the other networks in the afternoon (depending on your location). I don't have any information on the Sprint international plans, but if you plan on only using the phone sparingly for voice and text that will probably be a reasonably cost effective option.
There are a couple of important caveats to be aware of however....
- Be VERY CAREFUL with data roaming, as the costs can be HUGE. Were you planning on switching "off" the cellular data and only accessing the internet in Wi-Fi areas OR also using 3G/LTE data while you're out touring? It would be a good idea to discuss this with Sprint customer service, as you may want to opt for an international data plan too.
- While travelling with your home phone number, keep in mind that anyone calling your home number will reach you in Europe and YOU will pay for each incoming call (usually billed in one minute increments). Calls received in the wee hours of the morning from someone that didn't bother to check the time are especially annoying. Switching the phone "off" at night and using voice mail may also result in charges!
- Unlocking your phone and using a local SIM card may be slightly cheaper, but you'll be limited to using a single network, which could have coverage issues in some areas.