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, Asia 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 was known as one of the worst in the region for security checks and long delays, but this has improved markedly since 2017. 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.
Use a combination of flight search engines. Online booking an Uzbek Airways flight is possible nowadays. If it does not work out for you, check this page for timetables and online booking of domestic flights.
Since April 11th, 2017, regular flights are running between the 2 capitals. There were no flights for more than 20 years due to mistrust between the 2 countries.
Since November 2017, Tashkent is connected to Kabul through Kam Air.
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.