Q. What is the first mitzvah which a Bar-Mitzvah boy should observe?
This pattern was set at the time of creation when “there was evening and (then) there was morning” (Gen. 1).
Hence the first mitzvah for a Bar-Mitzvah to observe is the evening service (Ma’ariv). In particular it is the evening Shema that is his first adult act as a Jewish adult.
The Shema is the basic affirmation of Jewish identity. It contains the basic Jewish dogma (God exists and is unique) and the basic duty (to love God, study His Torah and keep His commandments).
Some synagogues require a Bar-Mitzvah boy to say publicly a specially written Bar-Mitzvah prayer: nice, but probably unnecessary, since the Bar-Mitzvah boy who has begun his adult career with the Shema has said it all.