Generally speaking, if you are in a professional situation you must give the other party a handshake. A handshake is done with your right hand and the other person’s right hand and involves joining them together for an extremely short time whilst moving them up and down.
The unwritten rule of a handshake is to go up and down twice, to keep it somewhat firm and to let go of the other person’s hand immediately after. You might seem a bit strange if you keep holding their hand.
In a casual scenario, it’s a bit different. You would seem odd shaking someone’s hand, although this is very culture specific. In a lot of countries, it is common to kiss the cheeks of people you are meeting for the first time.