Most hotels offer free Wi-Fi, though you still have to ask to make sure. Most shopping malls and cafés also offer free Wi-Fi, though to access it you usually need to provide a mobile phone number so the password can be texted to you. It is best to have a local number for this, but if you are using your American cell phone, just note that when you look for the country code, it is hidden under "American Samoa." Mobile data service is very good in Oman. SIM cards can be purchased on arrival with inexpensive data packages; mobile data works well on mobile phones throughout the country except in the deeper areas of the desert and in the Empty Quarter in Wusta.
The country code for Oman is 968. If you are calling within Oman—from an Omani number—no country code is needed, just dial the eight-digit number. If you are calling from outside the country or using a foreign number, you need to dial +968 (typically on iPhones) or 00968, depending on your phone type, followed by the eight-digit telephone number.
Calling within Oman
Phone numbers are eight digits long in Oman. Landline numbers begin with a 2 and cell phone numbers begin with a 9. There are no pay phones available.
Calling Outside Oman
International calling from Oman is very expensive unless you use a calling card. There is an inexpensive brand available in all petrol stations called "Aloo." Otherwise, most people use a VPN to allow data-based calling services like IMO, Skype, and Viber, which are all blocked. To call internationally from an Omani phone, dial "00" plus the country code (dial "001" for the U.S.) and then the number.
If you have an unlocked phone, you should definitely purchase a SIM card on arrival. You can get one for about 2 rials, and the full value of the card generally appears as a credit (more value can be added). Data subscriptions (1 GB) run about 5 rials and should last for your entire trip. You will not be able to rent a mobile in Oman, so if you need to rent an international mobile phone, do so in your country of origin. But if you bring an unlocked phone, it is cheap and easy to use an Omani SIM card. It is almost always cheaper to purchase an unlocked phone in your home country rather than purchasing one in Oman, where electronics are more expensive than in the U.S.