A strong handshake isn't a matter of squeezing somebody's paw.
It's a matter of presence.
On the street, in the lobby, square your shoulders to people you meet. Make a handshake matter — eye contact, good grip, elbow erring toward a right angle. Do not pump the hand, unless the other person is insistent on just that. Then pump the hell out of their hand. Smile. If you can't smile, you can't be gracious. You aren't some dopey English butler. You are you.
A handshake like that shows that you're paying respect to the person you're talking to,
and as science has confirmed, giving respect gets respect.