When I drill with a bit mounted in a chuck in the tailstock I noticed that sometimes the drill bits can wander off center as the bit starts cutting into the work piece. This is especially true with smaller bits. What I do is make a clean face on the end of the work piece with the lathe cutting tool then set the compound slider at a 30 degree angle. Then I position the lathe cutting tool to the center of the work piece and make a small dimple in the absolute center. That keeps the drill bits perfectly centered as I start to drill. Also if I'm using bits larger than 1/4" I will drill a pilot hole with a 1/16" to 1/8" bit.
If you're working with ferrous metals like steels then remember to use a lot cutting oils (cutting oil contains sulfer).
Also if you're drilling holes deeper than say 1/2" then back the bit out and to clean the shavings off the bit and out of the hole then add come cutting oil (if working with ferrous metals) then proceed to cut another 1/2" and repeat. This way you reduce friction from the shavings which creates heat and noise and helps reduce stress on the bit which makes it cut cleaner.
If you are making holes that require precision diameters like your bushings I recommend investing in some reamers too.
A lathe is a very versatile tool. The more you learn how to operate it you'll be amazed how much you can do with it. And since you have a milling machine attached to it I'd invest in a low profile rotary table.
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