: I read earlier that our whole solar system is rotating around our galaxy at like 250000000 miles per hour, so, if we were able to travel at the speed of light, you we were traveling at the speed of light, we would get a lot farther in the same time if we went against this spin instead of with it...

No, It does not say anything about effects of traveling at the speed of light.

: Does this mean that if a sun was traveling behind a planet and both were rotating, would the time it takes the light from the sun to get to the planet be longer than if the planet was behind the sun?

This second part is a different question completely. The answer is it depends on whose frame you go by. Simultaneity is relative not absolute. If I understand your question correctly, then the same physics can be understood in the following simplified situation. Accelerations due to rotation can be neglected taking this to be a locally inertial frame. And then it can be understood purely in terms of special relativity. Lets say there is a light emitting source in between two receivers an equal distance apart from the emitter. Now we will put two observers of events into the picture. One observer S will be still with respect to the emitter/receiver setup. The other observer G will be moving at some constant velocity with respect to the setup (Recall that this is a locally inertial frame). A light pulse goes off from the emitter. The two important events are the reception of the light pulse at each of the two receivers. Now the question asked would be does one receiver receive the light before the other, or do they both receive it at the same time. The answer is yes. It depends on whose frame you go by. There are two different observers with two different coordinate frames and two different standards for what events are simultaneous. According to the coordinate frame of observer S both receivers receive the light at the same time. According to the coordinate frame of observer G these two events occur at different times. This can be understood in terms of the first postulate of special relativity. The vacuum speed of light is locally invariant. For instance observer S may observe the distance of each receiver from the emitter is d and thus the time that goes by from emission to reception of the light is t = d/c for both. The observer G is using a different coordinate frame. G observes that the emitter/receiver distance is a different distance D and that the receivers were in motion between the time of emission and the two times of reception. Because of this motion and the speed of light also being c in observer G frame the times of reception are different.