What settings should I use on my Nikon D5100 for Taking Photos of Airplanes?

I'm going on holiday in a few days and I'm wondering what settings I can use to take pictures of planes, I don't think I will be using auto mode because it will be too blurry - I'm not sure though, I could be wrong(The planes that are moving).So If I were to use a different setting, what would I use (Aperture/Shutter priority - Or something Else), and also what particular setting is good for taking clear photos of planes whilst they are moving (Shutter speed).

I will also have two lenses with me - one is the standard nikon kit lens, the 18-55mm Nikkor and the other one is a 70-300mm Tamron lens, I will probably use the 70-300.Is that the right choice?

Refer to your handbook

As long as you keep the shutter speed at 1/500th second or faster, you will be able to capture aircraft without any blur caused by the aircraft's or camera movement. This can be done using either the manual mode or shutter priority mode on your fine camera

The 70-300mm lens will be the better choice.

This should help you:
http://digital-photography-school.com/a-beginners-to-capturing-motion-in-your-photography

The longer lens (70 to 300) will get you 'closer' to distant aircraft. But if the aricraft is close then you might need to use the 18 to 55. Use a reasonably high shutter speed - maybe 1/500, 1/1000, and use speed priority, if the aricraft is in action, not sitting parked on the ground. If the aircraft is seen against a bright sky you might need to use some exposure compensation, maybe 1 or even 2 stops. You could try using the Sport setting instead of Shutter Priority.