The Bazaar of Damascus is one of the main attractions in the city. The best-known part of it is the "bazaar tunnel" (a street with a ceiling) leading to Umayya Mosque. This is good place to observe local people and do shopping. You can see most amazing things, like Iranian women in burkhas buying pornoish underwear!
There are not many touts in the bazaar, so one can wander there without being disturbed with relative ease. But for sure, you will meet some salesmen who are willing to offer you a cup of tea. The salesman is hoping to sell you something, but you are not obliged to anything. The salesmen enjoy as well meeting tourists and having a talk.
I did some good shopping. Believe me or not: I bought some night dresses and sleeping gowns there! Both are extremely cheap and nice.