There are several sites around Tangier that are worth a peek. The Tangier American Legion Museum serves many purposes, including museum, cultural center, conference center and library.  Located in the old medina, it is the only historic landmark of the United States located abroad. The museum displays a variety of historical items and art exhibits, as well as the Paul Bowles Room, which is dedicated to the writer and composer who lived 52 years in Tangier. The library contains a comprehensive collection of books, monographs, maps, photographs and articles about Morocco and the rest of North Africa.

The Kasbah, which is the highest point in the city, is located within the medina but isolated by its walls. The gate opens onto a marble courtyard that leads to the Dar el-Makhzem, which was the former Sultan’s palace, built in the 17th century with carved wood ceiling. Today it is the Kasbah Museums, which includes the Museum of Moroccan Art and the Museum of Antiquities, and an archeological collection with exhibits primarily from Volubilis. Don’t forget to visit the Sultan’s gardens, as well. 

As any Moroccan town, the medina (old quarters) is almost a mandatory visit if you are interested in feeling "the life" of the city and being close to the locals (and other tourists too, for better or for worse). Perhaps not very "traditional", with much less of the "medieval" environment of other historical cities of Morocco, some of their buildings reminiscent of South Spain, others with a colonial look. Nevertheless, probably very exotic to the eyes of someone who has never been to non-western town before.

Perhaps you'll get disappointed if you are expecting to be impressed by this or that particular place you read about somewhere (guidebook, touristic panflet, magazine article, website, etc.). Don't expect much "pretty" sights, just try to breathe the environment and let go.

The area between the old Saint Andrews' Anglican Church, the Grand and Petit Socco and the Kasbah make a nice stroll and are easily walkable in 2 or 3 hours without too much hurry. But be prepared to some climbing, as the Kasbah relatively high.

Just up the Grand Socco, on the way to the Anglican church, there is a real traditional market which doesn't appear too popular with tourists, where you can find what appear to be real peasants selling there goods, dressed in there colourful traditional clothes. The women hats are particulary striking, very tall and decorated with coloured wool.

Another interesting place to "feel" the town it's the beach and it's avenue, which is fully crowded at the summer evenings. It seems that after dinner, every Moroccan goes out to the street!