Should I pray Fajr then immediately pray Subh, or should I wait until five o'clock? What is the difference between the two prayers, and how do I pray each of them?

The obligatory prayers are five. Fajr prayer is the same as Subh prayer. Its time begins with the true dawn (the horizontal whiteness in the horizon). When the time begins, one prays the two rak'ahs of the Fajr sunnah silently, then the two rak'ahs of the obligatory Fajr prayer audibly. As for the two adhans mentioned, they are likely the adhan before Fajr and the adhan for Fajr prayer, the latter of which is originally at the beginning of the prayer time. One can consult Islamic centers for further inquiries.