Arm Swing: Think of your arms as swinging on a pendulum. There is an imaginary rod sticking through your shoulders and your arms should swing back and forth on that. Do NOT cross your arms in front of your body. Try and keep them parallel to the ground. Your elbow should be at a ninety degree angle. Also, remember this phrase "the faster your arms move, the faster your legs move"
Hands: Keep your hands relaxed. Imagine that you are holding a potato chip in your fist and you can't let it fall out but you don't want it to break either.
Leg swing: This one is more complicated. A lot of people have a high back kick. That means their leg swings really far behind them (think butt kickers). This is very inefficient. Don't let your leg swing too far back. You also don't want to be doing high knees either. Find a good balance between front and back swing.
Push-off: DO NOT HEEL STRIKE. This is a very very common mistake many runners, even experienced runners make. When you land on your heels you are braking because it stops your forward motion. This also is a huge contributor to shin splints. With all that force going directly from your heel up through your leg, it causes a lot of stress to your shins. Try to land mid-foot.
Listen: Listen to your feet as they hit the ground. Does it sound like a pack of elephants or someone wearing flip-flops? Bad! This is usually fixed by not heel-striking (as mentioned above). Also think about running "light". It sounds funny, but if you try to make your steps quieter, your body will usually adjust.