Standing alone and touching each other’s hands is haram for a man and a woman if they are not married.

Since looking at a namahram woman is haram, touching her or shaking hands with her is definitely haram. Women who presented their obedience to the Prophet (PBUH) said: “O Allah’s Messenger! You did not hold our hands at the time of obeying you.” The Prophet (PBUH) said: “I never hold women’s hands nor shake hand swith them.” (Ahmad bin Hanbal, Nasâî and Ibn Majah). Hazrat Aisha says about obeying: “I swear by Allah that hands of Allah’s Messenger did not touch the hands of any women. He (PBUH) accepted their obedience only by words.” (Muslim).

In one of his Hadiths, the Prophet (PBUH) says: “Being poked with a needle in the head is more favorable than touching a namahram woman for any of you.” Islam does not underestimate women by the prohibition of shaking her hands, contrarily, it protects her honor, and prevents her from evil minded people touching her. (Halil GÜNENÇ, Islamic religious laws on today’s matters II. 112)

As all we know, considering a marriage with a girl and engaging to be married does not mean getting married. Therefore, going out or being alone with one’ fiancée is absolutely haram and a grievous mistake. The Prophet (PBUH) says: “When someone becomes alone with a woman, assuredly, the third one of them is devil.” When many engaged people become alone with their fiancées, some unwanted and illicit results occur, also their relations can be broken for some reasons, at the end. What is left after these relations, is sins and wrongdoings. Thus, people who take care of their religion, worldly lives and honor must be careful on such things.

Editor's Selected Link;

What are the Significant Points in the Relations Between Man and Woman?