As Peter does, I would recommend using a tapping-guide, and these can be used in the lathe (held in the tailstock chuck) in the same way as on a mill or bench drill.
I do find rotating a tap-wrench by hand on a lathe, physically slightly more awkward than on a milling- or drilling- machine, so raising the risk of breaking the tap. That is partly by more restricted access, but for me also due to being short, so the working position at my Myford lathe is not very comfortable unless I stand on a suitable platform. (Oddly, the much beefier Harrison is lower and more comfortable!)
When using the lathe tailstock as pusher for a tap or die, as Jason describes, I lock the tailstock’s position and gently wind the barrel forwards and backwards with its handwheel. For small diameters, by one finger on the handle. It’s more sensitive than trying to push the entire lump.
For tapping blocks and plates I’ve sometimes used the bench-drill as a tapper by slackening the belt and using the chuck as combined tap-holder / tap-wrench, manually. (Make sure the machine is switched off at the wall!). One hand rotates the chuck, the other gently eases the quill down and up – being careful to ease up when backing off the tap. The quill spring is quite powerful so I restrain the quill in this way, in both directions.