The lessons that I've learned so far was in 4/4 or 3/4. The tempo that I play lessons in 3/4 and 4/4 were between 62 and 72. The 6/8 lesson's tempo seems to be almost the double of the other ones. What's the reason for it? After researching about 6/8, I understood how the note values changes but the tempo change is still confusing me.
Your keyboard's built-in metronome ticks faster in 6/8 than 3/4 for the same tempo, because the metronome is probably ticking on eighth notes in 6/8 time and on quarter notes in 3/4 time, but tempo is specified as quarter note "beats" even for 6/8. There are two eighth notes for every quarter note, so you get two metronome ticks per beat. It could also tick once per every dotted quarter note i.e. 3/8 meaning twice per 6/8 bar, and you would then think of the meter as 3+3 / 8. There is no single correct way to specify what tempo "beats" mean in terms of your time signature, but usually it's beat = 1/4 note.