Does the direction you are traveling affect time dilation?

I see where you are coming from. I suspect you have got this idea from relative velocities e.g adding to a result one. However, for starters light doesn't work like that. If your in a car travelling at 50m/s and fire a bullet with velocity 1000m/s in front of you the bullet has velocity 1050m/s. Light is different. If you Are travelling at 0.5c and shine a torch in front of you then it light acted like sound you would expect it to travel at 1.5c. The speed of light is a universal constant though and can not be exceeded. The light would travel at 1.0c. Time will be dilated equally for both planets equally, provided they are travelling at the same speed. However, if you travel at 0.75c then your time dilation will be 1.51 (from special relativity equation. If both planets are stationary then your time will be 1.51 times slower than theirs. If however one plant is moving at 10000m/s through space then the time dilation for this planet will be 1.000000001. This is obviously negligible, since dilation only occurs exponentially close (above 0.75c) the speed of light. Hopefully although in a round about way, that does answer you question and allows you to understand why the answer is so.