You can immerse yourself into the Mongolian culture by spending some time in a traditional ger with a local nomadic family.
Mongolia is one of those countries that are less traveled and underrated but is worth traveling to. The capital city of Mongolia, Ulaanbaatar, is a wonderful place to disembark from the Trans-Siberian Express, and have a day or two to relax. After that you can start a new journey: it's easy to get to Beijing, China by train.
We also recommend you to explore the beautiful Gobi desert. Mongolia has the lowest population density in the world, and many Mongolian people still live in a nomadic way. You can immerse yourself into the culture, too, by spending some time in a traditional ger with a local nomadic family.