Unfortunately, there are no rail or subway services in Muscat, but two decent bus services are available. One, the "Baiza" bus service,  is inexpensive and will drop you off just about anywhere. In some cases, the interior leaves much to be desired, but air conditioning and timeliness make up for that. Bang on the roof when you're ready to get off, and the driver will stop. Another public bus service, recognized by its large red and green buses, is more expensive and limited to major roadways and specified dropoff points.

Taxis are run on a negotiable rate system. Expect dishonesty and abuse of unsuspecting tourists. Though the government attempted several times to institute the use of meters, the idea was repeatedly rejected. Only recently were taxis fitted with meters. Still, it's best to discuss the fare beforehand, just in case.

You may want to consider utilizing a travel agency for getting around in the city. They can set you up with a package to suit your interests and budget.

Paved, divided highways connect most areas, so road conditions are good. Traffic "roundabouts" are used instead of traffic lights; similar to rotaries. There's not a lot of car usage in Muscat, so these roundabouts serve quite well. 

A four-wheel drive vehicle is worth renting if you intend to explore the rugged country outside the city. You can either hire a guide or drive yourself.