When I'm learning the Qur'an by my heart I don't make ablution and I touch the Qur'an with my hand. Is that correct or not? Also I'm having my period and trying to memorize the Qur'an. Please help me to understand the best way

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Touching the Qur'an is unlawful for any person who is not pure. This is the opinion of the majority of the Muslim scholars. Their evidence is the following:
Allah Says (interpretation of meaning): {Which (that Book with Allâh) none can touch but the purified.}[56:79].
The Prophet also said: "Nobody should touch the Qur'an unless he is pure". So, you should be pure before touching the Qur'an.
As for reciting the Qur'an from memory, it is lawful even in the state of impurity. But if you need to touch the Qur'an in order to study it or to teach it while you are menstruating, you may use gloves or any other thing as to prevent direct contact, and provided what you are using as a partition to touch the Qur'an is pure itself.
Allah knows best.