This advice may seem basic.  The proper handshake should be taught when a child begins to interact socially.  You are the perfect coach. Look  the person in the eye and give a firm handshake.  As adults, I feel it is our job to teach this proper skill and demonstrate the skill.  What if we don't possess the skill ourselves?  

If you were overlooked and have a weak handshake it could give people the wrong or poor impression.  The best way to evaluate whether your handshake is up to par is to ask a safe friend or family member.  I am serious.  How do you know if you are giving the best first impression?

If I didn't encounter so many flimsy fish, half hand grips or iron grips, I would not be writing this.  Flimsy fish=weak; lack of self confidence, half hand grips from a  man to woman=the strength of a man is such that he doesn't want to injure the other party (women can handle it. believe me), iron grips=power trip, aggressive;  no eye contact=lack of self confidence.  Not your intention?  Then perhaps it is time to consider what kind of impressions you are making.

Make sure your hand is meeting the web of their hand, between the thumb and pointer finger.  Palms should be palm to palm and should not give more than 3 pumps.  Look the person in the eye and use their name as you smile and tell them it is nice to meet them. 

You will have them in the palm of your hand.