The main airlines that take you to/out of Mongolia are MIAT, Korean Air, Air China, Turkish Airlines and Aeroflot. Major cities where you can take these flights are Beijing, Berlin, Frankfurt, Seoul, Moscow, Tokyo, Bishkek and Hong Kong.
MIAT, Mongolia’s biggest airline company, flies from/to Beijing, Seoul, Tokyo, Hong Kong, Berlin and Moscow most of the year. Air China also operates flights to Ulaanbaatar from Beijing. Aeroflot flies from Moscow to Ulaanbaatar, and Korean Air flies to Ulaanbaatar from Seoul.
Trans-Mongolian train from both Moscow and Beijing might be the best choice if you are likely to travel by train. Train schedule is different depending on the season. In summer it is highly recommended to book your train ticket in advance as it is peak tourist season in Mongolia. There is a local train/bus that departs from Beijing to Erlian – a city on China/Mongolia border then cross border by van or take a bus to Zamiin-Uud, a Mongolian city and then take a local train to Ulaanbaatar.