As well as looking at the following information, you should know that a trip to Mongolia is not something to be undertaken lightly. The rules for entering the country change regularly. You must check up before you go.
The lastest information is that British passport holders only need a passport that is valid for 6 months after the end of their visit to organise a trip to Mongolia. You don't need to request a visa if your stay will not exceed 30 days. This information is very variable and is subject to changes and modifications. So it's best to find out from the Mongolian embassy before you plan to leave. If you want to stay longer than 30 days you must apply for a visa. It takes about 7 days to get one. You have to complete a form, and attach a passport-style photo, proof of your return journey and of your insurance. It also costs €60.
There are no mandatory vaccinations for going to Mongolia but you are strongly recommended to be vaccinated against typhoid, Hepatitis A, rabies, Meningitis A and C and encephalitis.
If you want to drive or rent a vehicle there you need to have an international driving licence. This is easily obtained at the Post Office. If you want to drive your own vehicle then make sure your visa mentions that you came over the border in a car. You will have to go the capital, Ulaanbaatar, and register with the customs officers. They will ask you to register your points of entry and exit from Mongolian territory, as well as the dates of your stay. Take your car registration documents with you as they will be needed.
If you leave Mongolia in the traditional fashion, by plane, you will have to pay an exit tax of $12.