I was wondering how a paintball ball is flying For example, does it fly straight, then up, then down?
This really depends on your barrel shape and the angle at which you fire.
The natural ballistic path of a paint ball (with nothing special done in the barrel), is to fly straight, and then down (really it's an arced trajectory that is the downward half of a parabolic curve). The downward motion is caused by gravity so the motion is governed by the basic ballistic equation. (d = vt-32.2t^2+h where h is the initial height in ft, t is time in seconds and v is initial (muzzle) velocity in fps and d is the distance in feet traveled at time t). This equation assumes you are shooting flat straight, however, if you are shooting up, then you need two factors in your velocity to account for the upwards and outward velocity, this can be determined using trigonometry.
More complicated barrel configurations may have rifling that causes the ball to spin coming out of the barrel causing it to have angular momentum that counteracts the initial effects of gravity and give it a longer straighter trajectory (this is mostly to counter the wind though, not necessarily gravity) Other barrels put backspin on the ball which causes it to have angular momentum against gravity again allowing for a longer trajectory in the ballistic arc. Both of these motion types introduce spin on the ball and complicate the equations governing the ball's fall in a way that is not easy to write out an equation for (the basic ballistic equation is basically universal, the equations for these would not be).
There are other factors that complicate the trajectory as well. The most notable factor is wind. If you are playing indoors, this is probably not a concern, but outdoors, especially in windy conditions can make shots very wild and must be compensated for.