Muscat is a very spread out city - more a collection of neighbourhoods linked by expressways. There's old Muscat - much changed these days by construction, but you can still find some old mansions with their characteristic studded wooden doors - Ruwi, a buzzy commercial district, and Muttrah with its harbour, fish market and souk.
I lived in Muscat for a few months and offer this selection of trips for the traveller;
* get properly lost in the souk. Most tourists only see the two main 'streets', but the souk actually contains numerous mosques, alleyways, restaurants and workshops off the beaten track. I particularly liked the silver trade shops with their selection of machinery and tools. Don't miss the halwa makers, stirring the glutinous date halwa in their huge cauldrons.
* Drive out to the Chedi, a fine zen-style hotel. The restaurant is expensive, but you get just the same experience if you visit the pizzeria - for much less!
* Go to one of the little Indian or Omani restaurants in Al-Khuwair. The Indian places provide a service for Indian expats, at a very modest price, and you can get a fine vegetarian thali or curry. The Omani restaurants are traditional, sit-on-the-floor places, where you'll get your own room and be able to taste some of the local cuisine (coriander and cardamom are the most typical flavours).
* Take a hike on the heights above Muttrah - make sure your shoes provide adequate support though, as the terrain can be rocky.
* Visit the fish market at Muttrah nice and early, and you'll see fishermen coming in with their catches.
* Take a walk or run on the beach at Qurum as the sun sets. Great for people watching - young Omanis playing football, fishermen coming in from the sea. From this beach you can see the backs of many of the Embassies and have a chance to compare the different styles of architecture, from po-mo to faux Arabian.