Anas (May Allah be pleased with him) reported: A man asked: "O
Messenger of Allah! When a man meets a brother or a friend, should he
bow to him?" He said, "No." The man asked whether he should embrace and kiss him? The Messenger of Allah (sallallahu 'alaihi wa sallam) replied, "No." He asked whether he should hold his
hand and shake it? The Messenger of Allah (sallallahu 'alaihi wa
sallam) replied, "Yes." [At-Tirmidhi]
This hadith prohibits greeting by bowing (a little or more) and embracing when it is accompanied by kissing the hand. There are ahadeeth that allow embracing alone when a person returns from a long journey or two people meet after a long time. For example,
On the authority of Anas who said: "When the Companions of the Prophet
(sallAllaahu alayhi wa sallam) used to meet they used to shake hands
and if they came from a journey they would hug." [Silsilah Ahadeeth
As-Saheehah No. 2647]
In general, the Prophet only asked to shake hands when the muslims meet. There is no reference or evidence for putting hands on chest or using two hands. The question put to the Prophet (salallahu alayhi wassalam) in the first hadeeth was about holding a single hand to which he replied in the affirmative.
So, the correct way is to just shake hands with a single hand i.e. the right hand because the Prophet (salallahu alayhi wassalam) used his right hand for all the affairs.