How can one begin studying the science of Hadith terminology, and what are the best books and methods for that?

It is not hidden from anyone with a sound mind that seeking knowledge of hadith is obligatory. Therefore, we start with a concise, gentle, and easy-to-understand summary that encompasses its rules, such as the book "Sharh al-Manzumah al-Bayquniyyah" by Allamah Muhammad ibn Salih al-Uthaymeen. Then we move on to "Nuzhat al-Nadhar Sharh Nukhbat al-Fikar" by Al-Hafiz Ibn Hajar, then "An-Nukat" by Al-Hafiz Ibn Hajar on the introduction of Ibn al-Salah. After that, we study books of Takhrij, such as "Nasb ar-Rayah fi Takhrij Ahadith al-Hidayah" by Az-Zayla'i, or "At-Talkhis al-Habir" by Al-Hafiz Ibn Hajar, and the introduction to "Tamam al-Minna" by Al-Albani. Then, we study parts of Al-Albani's two series, "As-Sahihah" and "Ad-Da'ifah." It is essential to memorize the rules, evidence, and sayings of the scholars, and to summarize the benefits for ease of review.