Here's how first time meets should probably go: eye contact, firm handshake, exchange of names, friendly smile, further conversation optional, end of story. It's really not that hard, though it can be awkward and nerve-wracking; but for some reason some people seem to not do one or more of these things.
Let's go through it one at a time, shall we?
1. Eye Contact
I'm surprised at how many people simply don't make eye contact. Personally, I find it somewhat rude when people don't look me in the eye. I know we're both nervous, please just do it.
2. Firm Handshake
Please no dead fish. Most men give me a firm handsake, but for some reason a lot of women don't think they need to. I think a nice, steady handshake shows that you care (whether it's business or casual) and that I mean business (when it's a business setting).
"Hi, I'm Sara." "Hi, I'm (fill in blank)." = How it should go.
"Hi, I'm Sara." "Hi." = Not how it should go.
This happened to me recently at work and for a while I didn't know the guy's name because he hadn't told me. It's not always a big deal if you're, say, never going to see the person again, or even if you are, you can just ask. But it always feels somewhat awkward, right? You feel like you forgot their name, even though they never told you.
4. Friendly Smile
Obviously you should be smiling the entire time you first meet someone to give the best impression, but an extra friendly smile at the end helps seal the deal, I think.
5. Further Conversation (optional)
Simple starters, like the weather or how the traffic was getting there or that you like the person's coat, etc., is always good, but not essential if it's just a meet-and-greet. Obviously, this part is pretty essential if this is a first date or if you're going to be working with the person.
Anything that bothers you about meeting people for the first time?