What is the best way to make up missed prayers: by making up a full day's prayers at a time, or by making up a corresponding prayer with each present obligatory prayer, and with which prayer should I begin making up if I do not remember from which obligatory prayer I started neglecting them?

Missed prayers should be made up in their proper order: Fajr, then Dhuhr, then Asr, then Maghrib, then Isha. One should strive to make them up to the best of their ability, alongside earning a livelihood for their family. If one does not know which prayer they started neglecting first, they may begin with Dhuhr or Fajr, and there is no harm in that. It is obligatory to be diligent in the future to perform prayers at their prescribed times.