They absolutely would not use a longbow. At least not the cavalry.

The advantage to the style of bow the Mongols used was that while it was labor intensive to make a composite bow, the results was a compact ranged weapon that didn't compromise in terms of physics. It retained a lot of the force a longer bow could deliver without being that big. A mongol bow was typically somewhere from half a person's height to 2/3rds.

Think about this for a moment- a longbow is traditionally as tall as it's user is. Do you really think someone wants to have that slung over their shoulder while they ride, let alone try and fire it? If the Mongols employed longbows it was by foot soldiers, not cavalry.

At the time of Ghengis Khan roughly 6 of every 10 cavalry was a specialized ranged combatant. Mongols were quite fond of the composite recurve bow (they used wood, horn and sinew lamination) which granted them accuracy and power quite a ways beyond what the size of the bow would suggest. Targeted shots could be made at 80 to 100 meters, while ballistic (not targeted so much as, "I Shot in this direction, lets see what I hit!") shots could be made at up to 400 meters. To supplement their bow, Mongols employed a wide variety of arrows, typically employing different designs for different targets, and different ranges. This often meant a cavalry archer could carry multiple quivers into battle.

Cavalry were divided into two roles- light cavalry and heavy. Compared to their contemporaries, Mongol light cavalry traveled very lightly. If you've played strategy video games, you're probably familiar with these units- Keshiks. These soldiers relied mainly on those bows I mentioned above, and once they'd done their business, they'd make way for the heavy cavalry.

Heavy cavalry rode into battle with lances of some sort and were intended for close combat, and clean up.

Typically soldiers also carried withe a scimitar, or similar thick-bladed sword, or an axe.

Whenever possible Mongols wore silk undergarments, mainly because they were remarkably capable of resisting attacks. While the skin may break, the silk wouldn't tear, making it easy to extract, say, an arrow head. Outside of that they would wear leather armor, and some lacquer armor- typically a leather shield of some form. They'd also wear a lot of wool. Shit gets cold in the Steppes and Siberian frontier.

Mongols were also expected to carry some simple kit in the field to repair equipment. An awl, some thread and needle, a knife, you get the idea.

After conquering parts of China, the Mongols brought in Chinese siege experts who introduced some siege weaponry- javelin throwers and catapults. Mongols would also later gain Balistae and were known to employ trebuche.

On campaign a cavalry soldier would have between one and five horses in reserve. For food they'd carry dried meats, fermented milk (not quite milk, not quite yogurt. You may have heard of Kefir, its kind of like that) and dried milk.

ah....and don't rely on wikipedia articles here. This article -> http://en.wikipedia.org/wiki/Mongol_military_tactics_and_organization is pretty poorly cited. They say these bows had an effective range of up to 1750 meters, and total range at 2000 meters, but then both offer no citation, and then say they had an effective range of somewhere from 100 to 400 meters.

Additional Reading:

http://www.coldsiberia.org/monmight.htm

http://www.mongolia-web.com/1203-mongol-military-tactics-and-organization