Nothing like a good old fashioned hand shake to say hello, show respect or cement a deal. Make sure you always shake with your right hand, even if you are a lefty, and get a decent grip. Don't try to impress the other person with a vice grip, but don't limp wrist it either, that's worse. A nice solid bite and brisk shake is all it takes.
If you are holding something in your right hand, then move it to your left and shake with your right. If you are at a party or social scene and you have a drink, make sure you use your left hand to drink so your right is free for shaking.
Do you always shake with your right hand? What kind of grip do you have?