Getting around Muscat might not be the easiest way to do however, there are a couple of ways in which you can do that. There are not railway or subway systems which makes it a little bit harder.
There are two bus services available in Muscat. One is the "Baiza" bus service which goes just about everywhere. It is pretty cheap and the buses are air-conditioned. When you want to get off you just need to bang on the roof of the bus and the driver will stop. The other bus service is the public one whose buses are large red and green buses. It is a bit more expensive than the other one and has specific stops.
There are different kinds of taxis that operate in Muscat. The first one are the maxi taxis (vans) which go past the highway from Seeb to the chronicle area. The price is fixed. You cn also use these taxis from the airport. The regular taxis are colored orange and white (mostly). They are a bit more expensive and you can find them around the bigger hotels. It is always best if you haggle with these kinds of taxis and agree on the price before setting off.
In Muscat, you will find a lot of rent-a-car services. It is not definite that you have to have an international driving licence, but some companies want it, so it would be best if you had on. Make sure that you have studied a map before setting off to somewhere because the road signs in Muscat can be confusing. If you are lost you can always ask for help from the locals.