Because of the large numbers of Uzbeks living abroad, and the many organised tours to the Silk Road cities, there are frequent direct flights to Western Europe and North America from Tashkent. Uzbek Airways is not a good airline, though. But it has a quasi-monopoly on flights to Uzbekistan, and a full monopoly on domestic flights. Tashkent airport is also known as one of the worst in the region for security checks and long delays. Almaty and Bishkek offer better options price-wise, although you will probably need to transfer. If you are doing a multi-country trip, consider starting and ending your trip there.
To find flights, we find the following flight search engines most useful: Dohop, Skyscanner and Kayak. Combine all 3 for a good overview. Online booking is not a reality yet at Uzbek Airways, so check this page for timetables and online booking of domestic flights.
Tashkent is connected by airplane with Andijon, Bukhara, Ferghana, Karshi, Namangan, Nukus, Samarkand, Tashkent, Termez and Urgench. Flights by Uzbekistan Airways are generally good value for money, but long waits, endless baggage checks and bureaucratic airport nonsense might persuade you to take the train.