That should do the job.
Spot drill 1/4, bore to 5/16, lubricate, tap.
Joey gave you the basics but given your level, you may need a little more of the in-depth specifics. After you learn what you're doing and why you're doing it, you can do it anyway you want.
Start by truing the shaft in the lathe and take a light face-cut to true the face to the center-line of the shaft.
Use a 7/16" center-drill to start your hole. You're going to want to go a little bit bigger than 3/8" on the dia. of the entrance to the hole so as to leave a small chamfer after the hole is tapped. This will be the true center of the shaft and that chamfer will allow you to always get back to center.
Apparently there are several size tap-drills you can use for the 3/8x10 tap.
In my work/job, I use a 5/16" drill because I don't know which style of 3/8x10 pin the shaft I'm building will go on. 5/16" hole will go on both.
Anyway, drill your hole to your preferred depth, usually 1/4-3/8" deeper than the length of the pin.
This might be a good opportunity to check your lathe. Once your hole is drilled, measure the id. The amount that it is over .3125" is twice the amount your tail-stock is off to the head-stock. You don't have to do this every hole. Just every once in a while to check.
I tap under power so this may not apply to you.
I use candle wax on the tap. Mark your tap so you know where to stop.
I tap to 1/2 depth under power and stop, manually back out the tap, clean out the hole, clean the tap & re-wax.
Manually start the tap back in the hole then finish tapping under power to depth (your mark) and stop. I then manually remove the tap from the hole.
Clean-out the hole, clean the tap, put your toys away. You're done.
Once you start doing ring-work, this is the time to do it, before you take the shaft out of the lathe. Everything so far is centered and true. Doing the rings now insures that they will be true to the hole as well.
Once you have the rings installed, you can now remove the shaft from the lathe and let the epoxy cure. For me, the shaft goes right into a bar-clamp to hold the rings in place and under light pressure.
Now your lathe is freed-up and you can start the next one.
After 8 hrs or so (preferably more) I can put my shaft back in the lathe using the live center in the tail-stock supporting the jnt. of the shaft via the chamfer I put in and trim the rings to the size I want.
It may be more than you wanted to hear but by the same token, there may be something there you can use.
Good Luck.