It is very common to see someone standing on the side of the road with their hand out, waiting for a random car to pull over and offer them a lift. Usually, the driver is willing to take you if you are going in the same direction as them, and when you arrive at your destination, you simply pay and get out! While in the USA it may be seen as dangerous to travel this way, in Mongolia, it is considered totally normal and is also very cheap.
To get from one city to another, people often take buses which are similar to Greyhound Buses, that drive to almost everywhere you would want to go in the country. These buses are not fancy, but they go to many places and are very inexpensive. You can also take the train, since Mongolia is part of the famous Trans-Mongolian Railway, which goes from Moscow, Russia, to Beijing China. This railway is one of the longest in the world, and to do it from start to finish would take more than one week!
When I first hailed a "taxi" in Ulaanbaatar, I was very nervous since in the U.S.A. I would never ask a stranger for a ride. But after I learned a little Mongolian language, I was able to give the driver directions and make sure I paid a fair price. Now it feels very normal to me to navigate in this way. When I took the bus, it was very crowded but much easier than I thought. I am used to taking buses in big cities in the U.S.A., but taking buses in foreign countries where I do not speak the language very well makes me a little nervous. I promised myself I would learn to overcome this anxiety, though, so I have been studying the bus routes here.
