Mongolia is fairly straightforward. The best way to obtain a visa is through your own embassy or consulate or in Hong Kong. Visas for British citizens cost ￡30. However, Mongolian visas can easily be obtained from the Mongolian consulate in Irkutsk (Russia). US citizens (90 days) and Israelis (30 days) do not need Mongolian visas.
Russia is more problematic. Invitations are generally required, and they must be registered in the country within 72 hours of arrival. However, Russian transit visas issued in Beijing or Harbin last 10 days and require no invitation. This would be enough time to make the trip (
K3,
K19) with no stops along the way and spend a couple of days in Moscow. The Beijing consulate is open from 9:00 to 11:00 but remember that many Chinese nationals are also trying to acquire visas with you, so show up early. The cost varies for each nationality, but Americans can expect to pay $250 for same-day service or $150 for the five-day service. Upon arrival in Moscow you have four nights valid on your transit visa, which allows for one or two nights in Moscow, an overnight train and one or two nights in St. Petersburg respectively, but you must be across the border before midnight on the final day of your visa.
There are many destinations from St. Petersburg, including buses to Helsinki, Tallinn, Riga, Kiev and various other places in Europe, but be wary that nearly all nationalities need a transit visa (or tourist visa) for Belarus so be sure to have the correct visa if your plans take you through Belarus.
It is generally assumed that border police stationed at bus routes that exit the country are less likely to make a fuss compared to the police on trains. A Russian transit visa cannot be extended under any circumstances. If you arrive from Beijing, you can register your visa after arriving in Moscow. If you have a 10 day Transit Visa and do not stay in one place (i.e go to St Petersburg) you do not have to register your visa, though it is recommended that you register your visa, if possible, within three days of entry into Russia.
Unfortunately, if you encounter police officers they might not have the same opinion and you could be faced with a "fine." Have your ticket ready as proof that you were unable to register sooner and keep all receipts from hotels and/or hostels from places where you have not registered. Israelis do not need Russian visas (90 days).