I have a rather long post somewhere in this forum on the topic, so I will be as brief as I can (I can hear the sighs of relief from everyone

)
Look, the holes (a) have to be square to the poppets and (b) have to line up. What I do is drill the headstock first and put in the center. Then I shove it over and use it as a center-punch for the tail. Then drill tail. You must have a really good square. When I drill, I tape a line level to the auger bit. I lay down the square to give me square and the level to give me the other dimension.
Now as to self-tapping. You can probably get away with this. But maybe not. Depends on size of tail center and your assortment of bits. There is a formula for this purpose. It is
drill size = outside diameter - pitch. (This gives you 65% coverage. Good enough for all practical purposes). So, for example, you have an M8/1 bolt (8mm outside diameter, 1mm pitch). Most 8mm bolts have 1mm pitch. So drill 7mm. Formula works in any units, but I refuse to deal with insane fractional inches. If you want inches go look it up on the internet, and be prepared to deal with Morse number drills. Google on "tap size," for instance.
And further, the
oh no! situation. So you do your best, and in doubt drill undersize. If the thing is too tight, go get a taper tap and run it through. Taps are not expensive. If it is too loose, all is not lost. Get a smallish piece of plastic. A piece of garbage bag will do. Roll it up tight around the bolt and screw it in. It will provide enough friction to run your tailstock properly. These remedies are based on personal experience, not hearsay. One of my lathes was too tight, and another too loose. I did as I advise.
I recommend a set of basic taps for anyone who builds stuff, lathes in particular. I have a set that goes M4-M12 and 4-40 to 1/2"-20 for inch fanatics. Cost $30 and I have never regretted it. I recommend taper taps. I can hear five generations of machinists howl

but taper taps are much easier on beginners than bottoming or finishing taps. If you have never tapped a hole before, there is a cardinal rule. Start the b---y thing dead square fore, aft, port and starboard. Use any means at your disposal to do so. A small square, or even two, is a great help. Two dimensions, you see. If you start the tap right you will have no problems. If you don't there is no fix for it.