There are 3 mammoth style mounts that allow more than one person on them and are available to both factions.
The first is the Traveler's Tundra Mammoth, a light brown colored mammoth and the easiest to obtain. It costs 20000 gold from Mei Francis in Dalaran which is in Northrend, a level 70-80 area but can be accessed at any level but is not recommended.
http://www.wowhead.com/item=44234/
The next most likely way you are going to get your mammoth is the Grand Black War Mammoth, which is a black mammoth as you you would expect from the name. It drops off any of the 4 bosses in the level 80 Vault of Archavon raid with a 1-2% drop rate per week each. This raid requires you have access to Wintergrasp and are at least level 80 to enter, but will not be able to be done alone at level 80. It easily done at level 90, the current level cap. Though this method is all on very thin luck, it is the cheapest and easiest.
http://www.wowhead.com/item=44083/
The final multi seater mount is the Grand Ice Mammoth, which is white in color. This is for sale by the Sons of Hodir quartermaster and costs 10000 gold (cheaper when your reputation with them is higher) and requires you to be exalted with them. This will take a long time (a week or two) and will still have a price (but not so expensive) but it is also not reliant on luck. If you are determined and patient, this would be the smartest and rarest of the 3 mounts to get.
http://www.wowhead.com/item=44086/
There are also single seater mammoths that are much cheaper/easier to get, like the black war mammoth which requires a measly 2000 honor points to purchase from the Wintergrasp quartermaster.
All of which requires the full game at least up to Wrath of the Lich King, which cannot be owned without Burning Crusade (which I think all comes free with the original game these days).
Good luck!