In some cultures they bow to each other, in some they rub noses, in some they just smile, but in Britain people shake hands when they are introduced or simply meeting up again.
It's easy. Just extend your right hand and take hold of the extended right hand of the person you are being introduced to. Shake it firmly but lightly and then withdraw the hand.
Clutching at the hand too quickly is to be avoided. It could alarm your opposite number, or worse result in your hand missing altogether and striking a karate-like blow to their groin. This would be a bad start to the relationship - unless of course the person is a black-belt and welcomes the unexpected practice (in which they will undoubtedly triumph).
It is best not to grasp with a vice-like grip. This can cause pain and may even result in a fight. Conversely, avoid the "wet-fish" handshake which implies that you are either weak, untrustworthy or sporting a poorly constructed prosthetic limb made of polythene jelly.
The actual shaking of the hand must be moderate. You do not wish to give the impression of suffering from some disease, or appearing to barely control an immense inner anger which might suddenly unleash itself on your new friend.
While shaking the hand you can smile and say "How do you do" or "Good to make your acquaintance". Don't however say this if you already know the person. If you accidentally do so, clutch at your chest, roll your eyes and say something to the effect of "These tablets are doing me no good at all".
Some word needs to be said about secret Masonic handshakes. These are special ritual greetings which should only be used by Masons. Attempting to gain social advantage by pretending to be Grand Master of the Lodge would result in having your tongue torn out by the roots, and your body buried in the rough sands of the sea at low-water mark. However you might just get away with a sound ticking off.
The concept of the good handshake is not difficult to grasp - smile, reach, clasp and shake. But remember to confine the shake to the hand - do not it extend to the entire body unless you are in a noisy night club.