How should I know when I should use the Cosine or Sine Law?

Generally speaking, if you know one pair of angle and its opposite side, then use the law of sine. Otherwise, use the law of cosine.

once you know two sides of an acute right triangle, you can readily solve for the other sides using the Sine or Cosine Law. Remember SOH-CAH-TOA for the trigonometric functions that will help you to solve. S-sine O-opposite side H-hypotenuse; C-cosine A-adjacent H-hypotenuse; T-tangent O-opposite A-adjacent

Sine Rule
a/SinA = b/SinB = c/SinC

For any given information you have take any pair of fractions.
The Sine Rule is best when the given information is two angles and one side, to find the corresponding 2nd side.

Cosine  Rule
a^2 = b^2 + c^2 - 2bcCosA

The Cosine Rule is best when the given information is two adjacent sides with an angle between them, to find the corresponding third side.

A variation on the Cosine  Rule is
CosA = (b^2 + c^2 - a^2) / 2bc

This variation of the Cosine Rule is best for three sides and no known angles, to find ,by inverting the Cosine function, the angle under the Cosine function.

NB Remember for any given triangle the capital letter 'A' , 'B' or 'C' refers to the angle. The small case letter 'a', 'b', or 'c' refers to the side OPPOSITE to the corresponding lettered angle.

Hope that helps!!!!!