Nukus is the bleak capital of the autonomous Karakalpak Republic, which comprises roughly the western 1/3 of the country. If you're heading to the Aral Sea or want to see Karakalpak culture, this is your best bet. However, Nukus has one extraordinary museum, that you have to check out if you are around.
The stunning Savitsky museum of art from Russia of the 1920s and 1930s - containing many thousands of items - is extraordinary and an extremely important collection of world-class Avant Garde art.
If you can find it, there is also a textile museum that highlights Karaqalpak cultural handicrafts and a walk-in yurt.
If you want to head to Moynaq, the former Aral Sea port, you're going to have to hire a taxi from the main bazaar. Expect to pay a lot because there is only one inconvenient bus and no one really wants to go there but tourists.