It's amazing how much "meaning" a lot of folks will implant into a handshake.
I used to be in a job where I had to shake a lot of hands, and learned a few lessons:
- Understand that it's an important social function to most people, even if you think it's pointless disgusting or useless. So by giving a good handshake, you're showing respect, and who wants to be dis-respectful toward your friends, acquaintances, bosses and colleagues?
- Since most introductions or meetings usually go in a predictable sequence, you'll have a mental split second or two to prepare - so you won't be surprised/flustered and will do a good job
- It's easy to mess up the initial "link-up" so I always mentally slow down and focus on looking at the other person's hand as I'm reaching, to ensure basically that we get the thumb/forefinger web of skin to hook up properly (key to a good handshake)
- The grip is well-discussed elsewhere. One thing about shaking lots of hands, everybody has different pressure requirements; but you can predict a lot by who it is - for example, nice old ladies don't usually have crushing grips, but type-A sales reps wearing "look-at-me" silk neckties... well, handshaking is a competitive sport with them. Be prepared.
- Also be sure to wrap your fingers around the others _hand_ and not their fingers.
- Then as soon as you've made good web-to-web contact and are easing into the pressure, look into your new acquaintances eyes, allow a friendly smile and say "nice to meet you" or whatever standard line you use.
Note that all of the above happens in a second or two... isn't the human mind amazing?