My favorite of Dubai proper was the older part of town.
The area includes the two oldest neighborhoods in Dubai…
…and between them the Dubai Creek, which has been used for shipping traffic since the beginning of the city’s existence.
On the north side of the Creek lie most of the souks (markets).
The Gold Souk, in particular, seems to be a popular draw for tourists and a handful of Emiratis.
The wharves on the north side of the Creek (conveniently near the souks) are packed with goods being loaded onto ships for point further into the Persian Gulf.
The south side of the Creek, called Bur Dubai, holds most of the traditional architecture.
Including the Dubai Fortress (turned Museum), and behind it the Grand Mosque. Taken together with their Creek-front location, the pair makes for a great place to watch the sun go down.
I’ve visited Dubai both as an independent traveler and while working as a photographer for GetYourGuide. Old Dubai and the nearby tourist sites are really easy to visit independently. If you’re more comfortable with a guide or bus tour, though, check out GetyourGuide for a number of options.