Just for the people saying the Mongols wouldn't want this: the Mongols
proposed it. The Europeans were the ones who didn't rise to the occasion. Also, just to be clear, 'Franco-Mongol' doesn't means 'France and the Mongols': just like the Muslim powers, the Mongols tended to refer to all Western Europeans as 'Franks'. So what we're talking about is the Crusading European powers concluding an alliance with the Mongols to wage a united two-front war against the Muslims.
That said: it's not exactly easy. The Mongols did offer it in OTL, but the Europeans were hesitant to ally with them, and the sheer distance would make effective co-operation rather tricky. Nevertheless, just by generally agreeing to attack from east and west at the same time in a roughly planned-out campaign would go a
long way, if you ask me.
As far as their goals are concerned, regardless of
@acrsome's dismissal, I'm afraid that "screw the Muslims" actually was pretty much it. For the Mongols, screwing over the islamic powers would allow them to push yet further west, or at least to greatly secure their western borders. For the Europeans, striking a crushing blow to the islamic world would pretty much ensure the success of the Crusading effort. And while they certainly understood that, after a defeat of the Muslims, the Mongols might just turn into the new big rival in the region, they also had to understand that the core of the Mongol domain was much further away. Basically, that would mean that the Mongols would have more trouble (and less inclination) to crush the Crusader states than the Muslims ever would.
All in all, a successful Franco-Mongol alliance would likely be a pretty hard blow for the islamic world, and a major victory for the Mongols and the Crusading powers. (Interesting side note: if the Mongols absorb a lot of Muslim land - and its population - that may simply ensure that the entire Mongol empire itself goes very islamic very soon. Which would be very interesting.)