At the Hazrat Imam Complex, you can visit the mosque, enter the library to see the world's oldest Koran, or sit in the shade and enjoy the view of the beautiful turquoise domes. For lunch, I recommend a local restaurant near the mosque, Karasaray Lagman. At that time, it was full of Uzbeks, which convinced us that it was worth stopping there for a meal. The restaurant's specialty was lagman, which is noodles with meat and vegetable sauce.
Our next stop was Amir Timur Square, named after an important figure in Uzbekistan's history. He was a ruler who conquered most of Central Asia, Iran, Iraq, India, and Transcaucasia during his reign, which is why Uzbeks still revere him today. Remembering him, they built a monument: Amir Timur sits on a horse and greets his subjects with his hand raised. Nearby, you will also find a museum dedicated to this figure. It features portraits of the ruler, models of buildings that were built during his time, and the scope of his conquests.
For a bit of festive entertainment, such as ice cream, shooting galleries, and even ping-pong tables, I suggest walking down Sayilgoh'chasi Street, from Amir Timur Square towards the Mustakillik fountains.