Buy a 3/8" tap, in my opinion it will save you a lot of headaches. I drill the hole, drizzle thin CA glue in it and let that DRY COMPLETELY. Then I tap the hole, and again drizzle thin CA glue on the threads. Then let it DRY again, and then clean the threads once more with my tap. I wax my the threads on my mandrel. 2 reasons, if you get the CA inside of the blank hot enough it will stick to the mandrel without wax, the other, it also makes it easier to remove the blank from the mandrel.
As to the drill bits, I believe the smaller one is for soft woods and larger for hardwoods. The softer the material the smaller you want the hole that you will be tapping.
I drill one day, add the CA, the next day I tap and add the second CA, third day I clean the threads and turn. Seems like it takes a lot of time, it does, but I would turn stoppers in groups of 24 or 36 at a time. I have a T Handle tap wrench that has the 3/8" tap in it, I made a homemade tapping guide that keeps it straight. All this is mounted on a 2x12 that sits in my lap, so I can tap as I watch TV.