Do the number of moons a planet has affect the speed of the planet's rotation? Would the tidal forces of multiple moons interacting upon a planet speed up or slow down the day length of the planet? I was thinking that maybe the planet's day length would be slowed down due to all of the gravitational tugging, but on the other hand I was thinking that maybe all of those tidal forces could perhaps slingshot the planet ahead and make it spin at a faster rate. Would the day length even change at all? If the answer is yes, could all of those moons even affect the planet's precession?
Any moon exerts a tidal force on its primary, tending to cause the primary to bulge outward along the axis passing the moon and to narrow slightly in the other two directions. This effect is greater if the moon in question is larger and/or closer to the primary, but a solid primary may largely resist it, so that it changes shape little if at all. In the case of the Earth and its Moon, the solid Earth moves relatively little, while the water of the oceans is moved quite a bit. As a result, water has to flow from one place to another over the solid crust. That flow causes friction, which causes the "bulge" of water to lag behind the Moon and pull back on it by gravity. This process gradually slows the rotation of the Earth and drives the Moon further away.  

This process acts, in general to bring the rotation of the planet and the revolution of the Moon to the same rate, so that the planet keeps one face to the Moon. Usually this means slowing the planet's rotation, but a few moons are so close to the planet that it would be the opposite. This process will be less pronounced on a planet which  has a deeper liquid layer, since there will be less friction, or on a planet which has a stiff solid structure, since there will be less movement. 

The number of moons is not directly relevant. To a reasonable approximation they all work independently.