In addition to the Vladivostok-Moscow train going through the entire Trans-Siberia Railway, there is also the China Railways K3/4 train which goes from Beijing to Moscow through Mongolia. After departing Mongolia, the train enters the Trans-Siberian Railway at Ulan-Ude, Buryatia, Russia, and follows the same route as the Vladivostok-Moscow train.
You can thus (almost) cross Asia southeast to northwest, beginning in Hong Kong (or Hanoi, if you prefer). Take the Z98 train from Hung Hom Station, Hong Kong, and you'll get to Beijing West Station. Then take the aforementioned train to Yekaterinburg (requires a change of stations, but nothing too bad), right to the east of the Ural Mountains, which is the Asia-Europe border. If you prefer to start in Hanoi, take the China Railways train number Z6 at Ga Gia Lim, Hanoi, and you'll arrive in Beijing West Station after 3 days. The soft sleepers (which I recommend; they are pretty nice) on the Hong Kong to Beijing train costs around 950 RMB (~137 USD) per ticket, and if you want something cheaper you could go for the hard sleepers at around 750 RMB (~108 USD). I don't know about the Hanoi to Beijing train, but it can't be much more expensive than this.
Unfortunately, there is no passenger rail transportation in Laos and very limited passenger rail in Cambodia, so you can't go farther south. If you begin in Singapore, you could get as far as at the Thai-Lao border on rails. If you manage to go across Laos and into Vietnam by other means, you'll eventually get to Hanoi, where you could hop on a train to Beijing (and eventually Yekaterinburg).
There are two ways to cross the North American continent from East to West within the United States. You either: take the Cresent Train from New York City, Philadelphia or Atlanta to New Orleans, and continue on the Sunset Limited to Los Angeles; or: take the Lakeshore Limited or the Cardinal from New York City or Washington, DC, then continue on any one of the Empire Builder, the California Zephyr, the Southwest Chief, or the Texas Eagle trains to Los Angeles.
There's also this luxurious Rovos Rail from Cape Town, South Africa to Dar es Salaam, Tanzania. In principle, you should then be able to take a few train trips to get to Alexandria, Egypt on the northern rim of Africa (via Addis Ababa, Aswan, Cairo, etc). In practice, I haven't found viable train route yet.