Yekaterinburg’s airport is located 15km from the city centre, and is easily reached via the #1 bus. One of the busiest airports in Russia outside of Moscow and St. Petersburg, it is used as a hub by a number of airlines and services over four million people every year. There are three daily flights to Moscow, along with daily runs to St. Petersburg, Samara, Irkutsk and many others. There are also plenty of flights to the usual European hubs (London, Frankfurt etc).
The main train station is just north of the city centre, and Yekaterinburg’s status as a major regional city means it is no surprise that there are plenty of daily connections to the country’s other big cities. That is a fairly liberal use of the word ‘daily’ of course, as Moscow (25 hours) and St. Petersburg (37) are over a day away. Trains also head to Kazan, Novosibirsk and more on a daily basis.
There are two major bus stations in the city, but neither are particularly reliable or useful. Neither has much going on in the way of facilities, and the timetables change more often than timetables should. Best to stick to train and plane for your transport needs.