For local culture, it's definitely worth a look around the Mutrah souk. While it's got tourists, especially near the entrance, one must bear in mind that Oman, and Muscat in particular, has a culture with a long trading history. Selling stuff to visitors is not a new activity for modern tourism, it's an ancient activity in those parts.

Go beyond the entrance and the first few passageways to look around; you can get most any portable house or food goods there. Down on the second left or so there is a shop selling a fabulous array of gas burner cooking rings There are many places selling shawls, as well as other clothing. Obviously plenty of people selling frankincense, and other spices. If you want to look like you know what you're doing, ask for green frankincense - this is the highest grade, not usually on display, meant to be eaten or burned. The lower grades are for burning as incense.

Bearing right from the main entrance, or in its own entrance a little to the west of the main entrance is the Gold Souk and the related Silver Souk. Gold is ridiculously overpriced these days so only buy if you want it; the silver goods are more finely made and better priced. Goods are sold by weight, the amount of work (detail), and how much profit the seller thinks they can make. Polite bargaining can get you a discount, I tend to get 10-30% off.

If you don't want to go far to eat from the souk, the second schwama-and-juice place to the right as you leave the main souk entrance (towards the nearby fort) is tourist-priced but pretty tasty.

The Sultan Qaboos mosque is also worth a visit, especially if you haven't been to a lot of mosques: it's quite traditionally styled and very completely finished and decorated. It's only open to non-Muslims in the morning (to about 12:30 I think) and you will have to wear Islamically-modest clothing: trousers and longer sleeved shirts for men, long sleeved top and long skirt or trousers for women, headscarf for women. The custodians are polite but the dress code is enforced. They have loan clothing in the form of a general cover-all if you are deemed to be unsuitably dressed.