Did you know when it comes to a handshake, if you don't make eye contact, you're immediately seen as less confident? I learned this is one of my professional training seminars. It is easy to be nervous walking into a social situation and look down to make sure you don't miss a person's hand. This makes you appear uncertain and unsure. In actuality, you can see your hands out of your peripheral, so you would rarely miss, and can thus maintain eye contact and exude the aforementioned confidence.
Of course, we're not talking excessive staring. Since you will most likely be talking with someone, saying things like 'good to see you again' or 'nice to meet you, I am _____', you'll be walking into a situation where natural speak should be coupled with strong contact to demonstrate self-worth and power.
We could talk about cultural customs or whose hand goes on top of whom at a later time, but for now, we'll stick with the traditional shake:
As your hands come towards one another, the space between your thumb and index finger will touch theirs, and your fingers will wrap around the palm and lay on the back or side of the hand. Grasp strongly.
Some people suggest squeezing only as hard as the other person if you are in the less powerful position or having a firm shake if you are displaying dominance.
Remember, at the end of the day, it is just a handshake. Some people don't even prefer them at all. As long as you are being yourself and being confident in how your respond and speak, you will be just fine.