Bill Raby has videos on YouTube that show building a rifle that are absolutely terrific. His videos are really worthy to watch and he takes his time doing what I think is really good work.
I want to address drilling speed, you are doing one hole, you don't need production speeds and flood coolant. Make the barrel secured so it will not move as has already been said, but it is really important to get a good job done. Set the speed to a low setting like 130RPM, not screaming fast. Let the drill bit make nice easy cuts and don't force it to the point you are generating smoke and heat. You can use a cutting fluid, WD 40, kerosene, liquid soap or similar. You can also blow a stream of air on it. It cools the cut and flushes away the chips. You can drill it without fluid, cooling , whatever but keep the chips clear.
If you drill a little, pull back, drill a little, it will make a nice hole and you are less likely to screw up.
Have control of the drill feed so it doesn't grab the work and suck itself into it. So a drill press is great for that. If the work isn't held down and the drill grabs the cut, it can hurt you, break the drill bit, or make you regroup.
Here is a picture of a tap, tap handle, and spring loaded tap guide
The tap guide shown at the bottom is 1/2" diameter into a chuck which in your instance is the drill press chuck. The pointed end is spring loaded and the point fits into the centering hole in the tap handle. When you clamp your rifle barrel to the drill press table, the tap follower is in the chuck and is in alignment with where you want the hole in the barrel. The tap follower point is spring loaded and can be pushed in against the spring (one shown will move 1/2"). Set the distance from the chuck to the table so the tap follower, Tap handle with tap, and the barrel are in alignment and the tap follower is compressed enough to hold everything together. You want the tap follower to always have the spring pressure on the tap so it stays lined up as to turn the tap handle. Same thing when you are backing the tap out, don't allow the tap follower to bottom out compressed as you remove the tap.
It won't hurt a thing to practice doing the setup on a piece of scrap metal so you will know what to do on the barrel.
When tapping metal I really like the stuff shown in the bottle.