Approximately, what is the number of authentic Prophetic narrations (hadiths), and how can a non-specialized Muslim read and learn about them? Is it better to start by reading and understanding their meanings or by memorizing them?

The authentic hadiths narrated from the Prophet, peace and blessings be upon him, cannot be confined to a specific number, and any figures mentioned are approximate. Benefiting from authentic hadiths does not depend on enumerating them. The most esteemed books of authentic hadiths are Sahih al-Bukhari and Sahih Muslim, in addition to what has been authenticated by the حافظ (hafiz) imams, such as Imam Ahmad, Sufyan al-Thawri, Ibn Uyaynah, and Shu'bah, and among contemporary scholars, al-Albani and Ahmad Shakir. Engaging in reading hadiths and gaining a deep understanding of them, or memorizing them, are both good. However, deep understanding and comprehension are superior to extensive memorization without understanding.