Oman Air (http://www.omanair.aero) is the national carrier and flies regularly between the two airports in the country Muscat/Seeb, and Salalah. Air Arabia now offers flights to Salalah and Muscat from the United Arab Emirates UAE.
All Taxi drivers in Oman are Omani nationals as this is a protected profession. In Muscat there are call/telephone Taxi services. Whilst safe and generally turn up when you want them to the costs are comparatively high. Look for "Hello Taxi" and "Muscat Taxi" amongst others.
The orange-badged taxis are usually owner-operated, these are unmetered with negotiated fares before departure. If you get a very cheap price, then do not be surprised if the Taxi stops to add extra passengers unless you request for it to be private. You may ask for engaged, just say 'engaged taxi' to the driver, and you will pay for all the seats 4 and now have the taxi to yourself. Women must always sit alone in the back. This is for your own safety and comfort.
There are also mini-buses Baisa buses, the principle is you share the bus or car with others and pay a lower price as a result. This is how women living in Oman travel if they must use public transport. Women should sit next to other women if there are any in the bus. Men should move to other seats. If they do not move immediately, simply stand at the door, looking at them expectantly. They will take the hint and move. Although this might feel strange to foreigners, it is expected behavior for Omanis. Not sitting next to a man will avoid any unfortunate situations of mixed signals.
There are regular, daily bus services connecting the bigger cities within Oman Muscat, Salalah, Sohar, Sur and Nizwa. There are several, daily bus services from Muscat to Dubai. There is one bus a day from Muscat to Abu Dhabi. For details see the pages of the Oman National Transport Company (http://www.ontcoman.com).