The best way to experience Damascus on ones first day is to hire a guide. The city can be confusing to navigate and English is not spoken pervasively in town. The Four Seasons arranged an excellent guide for us, and we visited the National Museum (where there are many artifacts from old Damascus, said to be the worlds most ancient surviving city). The museum also boasts an amazing ancient synagogue that was unearthed by Yale University researchers excavating in eastern Syria. We later visited the old train station (its getting rebuilt), which was once on the Orient Express line on the way to Baghdad.
Finally, we toured the Old City, walking through the famous souks (all the different types selling spices, foods, leather goods, CDs, jewelry, and more), jostling about the crowds and taking in the energy of the market. We visited the Umayyad Mosque (or Grand Mosque, which draws pilgrims from all over the world), stopping briefly by a church (about 10% of Syrians are Christian). Our guide also pointed out some of the famous hammams. Men, try the Nour Eddin Al Shaheer hammam for an elegant setting (men only except for Fridays, when its women-only). Women can try Hammam Al Bakri. A hammam is a great way to wind down and cool off after a long day of walking. I have always wondered how a hot steam can, upon exiting a hammam, cool you off on a hot summer day!
For some bar action, try Oxygen (a restaurant, too) in the Old City, or Marmar (which has a younger, trendy set). I was told Z Bar, located on the rooftop of the Omayad Hotel, is also a fun spot. More recently on the scene are O Lounge (an outdoor summer spot) and Dome, a 14th-century hammam-turned-restaurant/bar located outside the city. The latter two are owned by a mix of Lebanese and Syrian partners; the idea, with both spots, is to replicate the ambience of famous Beirut-style clubs like White or Sky Bar. The result, in O Lounges case, is a sophisticated atmosphere perfect for a late evening escape.
I certainly would enjoy returning to Syria for a longer stay. I learned that other important tourist sites worth a visit, include Palmyra and Basra (for their ancient ruins, including the particularly moving Roman coliseum in Basra) and Maaloula (a scenic Syrian village). The ideal itinerary would include several days in Damascus and then day trips to these other sites.