Most travelers will need visas for all three countries.
China and Mongolia are 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), and Chinese visas in Ulaanbaatar(Note: For the moment it is not recommended to apply for Chinese visas in Mongolia, due to tightened regulations.) Americans (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 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 exits 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 (see here if unsure) so be sure to be prepared with a 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 versus 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 Saint Petersburg) you do not have to register your visa. 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've been unable to register sooner and keep all receipts from hotels and/or hostels from places where you haven't registered. Israelis do not need Russian visas (90 days).
通常你可以在俄罗斯任意车站买到，不一定要在其铁路沿线的车站，有时这样甚至更便宜。价格的差别不是在于车票本身，而是是否包含保险。你可以自己选择要不要买保险：说bez strahovki (без страховки，不要保险)或者so strahovkoi (со страховкой，要保险)。车票印在一张双层的橙色纸上，保险印在一张同样大小的粉红色纸上。
The Trans-Siberian trains have varied schedules - some trains are daily while some go on even dates, some on odd dates and some trains depart only on a couple of days during a week. There are also passing-by trains (проходящие поезда), which are actually legs of longer train itineraries. E.g. a Ekaterinburg-Irkutsk leg of a Moscow-Vladivostok train. In this case not only schedule, but also availability is affected - such tickets are released for sale 24-48 hours before departure.
Russian Railways has all Russian train schedules, as well as some of the international trains departing from Russian destinations (e.g. Moscow - Beijing train). Only actual availability is shown, which is released 45 days prior to departure for all Russian trains except for the passing-by ones and 30 days for most international trains. You will need to use alternate spellings for some destinations. Beijing is called Pekin, Moscow is Moskva, Saint Petersburg is Sankt-Peterburg, Yekaterinburg is Ekaterinburg or Sverdlovsk (old name of the city), Ulan Ude is Ulan-ude, Ulaanbaatar is Ulan-Bator, and Khabarovsk is Habarovsk.
Coming from Beijing or Harbin, the last stop in China is Manzhouli. The food being sold there is quite expensive, but many Russians stock up on provisions (i.e. spirits and beer). Be aware that you can take a maximum of five beers (Harbin Beer, 0.3l) per person into Russia or you will have to pay a penalty (read: bakshish) to the customs. Get rid of all your Chinese Yuan here as they become virtually worthless once abroad, unless you want to take them as a souvenir. There are a couple of black market money changers in front of the station that change RMB to Roubles at rip-off rates. To get Roubles you have plenty of time on the Russian side of the border (Zhabaikalsk). Walk to the ATM located at the bank in town. Allow 30 minutes to go and come back. The train stops for several hours while the carriages are being changed, so you can do some shopping at the local food markets (bread, cheese, etc.).
Coming from Beijing via Mongolia into Russia there are still the same rip-off exchange touts, but most if not all platform vendors in Mongolia and Russia take US Dollars or Euros. However, they only take bills (or notes), so know the exchange rate and buy a lot if you are using a five Euro note. Always ask the attendant how much time is available before you rush off into a station to find a Bankomat (ATM) because the train will not wait for you. If you are not spending time in Mongolia, don't worry about acquiring Mongolian tögrög. They are worthless virtually everywhere else, and the export of tögrög is technically forbidden. Therefore, spend Dollars or Euro, but get Roubles ASAP because Russian vendors are more likely to fabricate exchange rates than Mongolian or Chinese platform vendors. Dont forget to buy a lot of vodka while in Russia!
In every train car there is a pot with boiling water available for making hot drinks (bring your own tea, but the water is free). Carriage attendants also sell tea and coffee, and it's usually possible to buy soft drinks, beer and vodka in the restaurant carriage to bring back to your carriage.
All tickets for long journey trains are for sleeping places. Trains between Moscow and St. Petersburg have seating places. Most trains in Russia have 3 classes of cabins to choose from;
Note that sometimes there is no shower in the train. Even in the 1st class on K19 (Trans-Manchurian). You can have an Asian-style hot shower though, if you bring along 2 jars. Fill one up at the hot water dispenser, go to the washroom and mix the water you get there in the second one.