It depends on his charisma. With a Charisma of 17, let's say, he gets 1 bonus 1st level spell (from page 8). So at 2nd level he can cast his Spells Per Day for the Class + Bonus Spells. So for a Bard, it is 0 + 1 = 1. He can cast 1 1st level spell per day.
At 3rd level, a Bard can cast 1 1st...