How does light actually behave? Senario:
I am walking 5 meters/second and have a laser pointer that I shoot a burst of light from perpendicular to to the path I am walking on.
Would the light travel perpendicular to the path from where I shot it or would it travel "diagonally" because I am walking 5 m/s perpendicular to the direction I shot it? 
 A: If you travel with $v$ = 5 m/s in the x direction and shoot the photon with $c$ in the y direction it will have $\sqrt{c^2-v^2}$ in the y direction and 5 m/s in the x direction (Pythagoras) for an observer which is at rest relative to the ground you are walking on.

Since you have 5 m/s relative to this observer he will find that the photon moves with $\sqrt{c^2-v^2}$ relative to you (which is less than c), because the x-axis cancels out
In your own initial system you are at rest and the ground is travelling relative to you. The photon then has a velocity of exact $1c$ on the y-axis, since you are always at x=0, y=0.
A: You can figure out the correct answer using the two fundamental postulates of special relativity. The first postulate tells you that if the beam travels parallel to the body of the laser pointer in the ground frame when the pointer is at rest relative to the ground, then if someone in another frame has a laser pointer at rest in their frame, the laser must travel parallel to the body of the pointer in that frame. So, if a laser pointer at rest relative to the ground and aimed in a vertical direction will shoot photons out that move in a completely vertical direction in the rest frame of the ground, it must likewise be true that if a laser pointer is held by an observer moving 5 m/s horizontally relative to the ground and aimed in a vertical direction, the photons will shoot out in a totally vertical direction in the rest frame of this observer. And if each photon is traveling totally vertically in the rest frame of an observer moving 5 m/s horizontally relative to the ground, this tells you that in the ground frame each photon is moving 5 m/s horizontally, and the second postulate tells you the total speed of each photon must be equal to c in all frames (299792458 m/s), so as Симон Тыран said you can use the Pythagorean theorem to determine the vertical velocity, since (total velocity)^2 = (horizontal velocity)^2 + (vertical velocity)^2. Thus, the vertical velocity of each photon in the ground frame must be $\sqrt{299792458^2 - 5^2} = 299792457.9999999583045$ m/s. 
