As anywhere in China, Mandarin is the lingua franca; nearly everyone can speak it. The Mandarin spoken in this region, especially around Beijing, is the standard dialect. Inner Mongolia naturally has many speakers of Mongolian, but many of them speak Mandarin as well. Russian was fairly common as a second language in the Northeast during the Soviet Era and even before that. As elsewhere in China, English is not widespread but some people speak it quite well.
As elsewhere in China, there is an extensive rail network. Rail is the main means of inter-city travel for the Chinese themselves, and many visitors travel that way as well. The system now includes fast bullet trains on most major routes; unless your budget is very tight, these are the best way to go — fast, clean and comfortable.
All the major cities have airports with good domestic connections; some have international connections as well. See the individual city articles for details.
There is also an extensive highway network, much of it very good. Busses go almost anywhere, somewhat cheaper than the trains. See the China article for more. Driving yourself is also possible, but often problematic; see Driving in China.