If calling from overseas, the country code for Zimbabwe is 263, but drop the initial zero for area codes. The international access code from within Zimbabwe is 00. International calls can be made from your hotel or lodge.
Easily the best option for making calls is to purchase a prepaid SIM card (US$1), available on arrival at the airport; these are cheap and easy to arrange. Econet and Telecel are the main operators and have branches throughout the main towns, as well as at Harare airport. Econet has the best phone and data coverage across the country, but in national parks you won't get network. Be aware that your phone needs to be unlocked to activate the sim. Credit or 'airtime' is widely available from street vendors.
Mobile numbers are recognised by the 07 prefix.