We used the popular app Whatsapp during our most recent trip to Italy. It proved to be very helpful as it allowed us to communicate with family and friends at home without incurring any international charges or utilizing any data. All we needed was WiFi, which was free and readily available in most public locations, and in our private accommodation, and for the other party to download the app onto their phone as well. It can be downloaded on most phones (Android, iPhone, Windows), and also features a handy group chat that has allowed our travel group to remain in contact since our return.
The fact that Android users like ourselves can use it to communicate with iPhone users, makes it worth it. It's just like placing a call at home, and there is very little lag time despite the call being placed over the internet.
A tidbit is to make sure your phone remains on airplane mode the entire time during your international stay and turn on your WiFi manually. This is more common for short-term travel where internet is readily available versus long-term travel whereas it's advisable to purchase either an international plan (AT&T international plans have a minimum of 30 days) or an international chip for your phone. Another suggestion is to switch to T-mobile (if you aren't already with them) which allows for all calls made internationally to be free.
Of course email is still a popular way to keep in touch, as well as social media sites like Facebook and Instagram.
Have you tried Whatsapp? Are there any other communication apps that you use to keep in touch with family when traveling internationally?