Continuing, carrots cooked and eaten.
Bits with a proper centre help, like a wood bit, forstner or spade. A small c 4 mm pilot hole might be useful if you are using a general purpose bit.
Lathe speed is a bit "see how it goes". If you can smell burning it's too fast
. If it's cutting well and small you can crank it up a bit, I usually start around 500 rpm. Assuming you have a well marked centre, with a bit of a hole (bradawl or nail) to stop it wandering from the start, bring up the tailstock, lock it and start drilling by turning the handwheel as you normally do Establish the hole, maybe 30 mm or so, then unlock the tailstock and withdraw the drill by pulling back, left hand on the Jacobs chuck but nowhere near the spinning wood. That clears the flutes much more quickly than winding it back on the handwheel. If it's all cutting nicely all you need to do is keep sliding the tailstock in to drill and out to clear , if it's hard going, type of wood or drill not great, then you bring it in until you get contact and go back to using the handwheel. The main thing is not to be afraid to bring the tailstock back to clear debris. If your first 30 mm or so is where you want it the drill bit will always find it's way back in the right place.
For this to work, the tailstock needs to move freely so you can just ease off the lock rather than loosen it too much. Assuming you have a cast bed lathe, clean the bed and rub a little beeswax block or similar along it. Don't forget the underneath, work out which bits of the casting the tailstock locking plate runs along, clean and wax that as well. Good general practice anyway regardless of drilling. (Makes the banjo move nicely). No experience with bars but I guess there will be some suitable lubricant.
Think what the hole is for. When I do a table lamp drilled from both ends I know there is zero chance of 8 mm holes meeting in the middle, they drift. I do 8 mm from the top then 25 ish from the bottom, no one will ever see that the 8 is off centre when it meets the 25, all that matters is feeding a cable through.