What is the ruling on speaking to a young man and getting to know him, going out with him, allowing him to kiss me, then swearing on the Qur'an not to return to that, and then resuming contact with him again?

The questioner committed several transgressions, including: not remaining firm in her rejection of this young man and opening the door for communication with him, allowing him to kiss her, which is a sin, betraying her promise to her family to stay away from him, and breaking her oath after swearing to them. These sins require immediate repentance, as relationships between genders outside the framework of legitimate marriage are forbidden and lead to corruption. Speech between a non-mahram man and woman is not permissible except with legitimate Islamic restrictions, and the relationship with this young man must be severed immediately.

As for swearing on the Qur’an, it is an oath just like swearing by Allah, and the Qur’an is the word of Allah and one of His attributes. Therefore, swearing by it is a binding oath that requires expiation if broken. Since the questioner broke her oath, expiation is required of her. If the young man is serious about marriage, he should propose to her through her family.