Buses are the cheapest way to get around in Muscat. Modern buses travel major roads with specific bus stops. For more out of the way destinations, Baiza buses are common and zigzag through the back roads effectively, although the buses themselves are sometimes a bit dilapidated.
Taxis are widely available and an easy way to get to and from the airport. Insist that the driver uses their meter unless there isn't one, in which case agree on a price before getting into the car. Taxis are expensive but convenient when you can't find a bus or don't want to wait in the sun. There is no subway or railway in Muscat and some travellers decide to rent a car and drive themselves around, which is the best option for flexible travel.