If the lathe is long enough, you can guide a hand reamer with a centre at the "handle" (Square ) end and let the cutting end find its own way, while you hold against rotation with a Tap Wrench.. In that way it can follow the drilling, without any side thrust.
You use the Tailstock barrel to provide the infeed, as well as steadying it. Don't rush it, but equally, don't let it rub.
Having several helical cutting edges, it should produce a good finish.
If a tool rubs rather than cutting, it may overheat at the cutting edge, soften and become blunt.
Finally, if you are new to lathework, it is worth spending some money on reading matter.
It took me a time to learn that putting on a cut removes twice as much as I thought, until I realised that it came off both sides of the bar! My turning instructor let me find out that for myself!
A set of Zeus charts will be invaluable.
The one or more:
L H Sparey "The Amateur's Lathe"
Ian Bradley "The Amateur's Workshop" Includes detail on setting up the lathe to minimise any twist. (If a lathe bed is twisted, it will not cut parallel )
Harold Hall "Lathework"
Both Neil Wyatt and Dave Fenner have each written books on "The mini lathe"
Tubal Cain "The Model Engineer's Manual" is very useful reference book, with lots of details on many subjects.
As you gain experience, so will your confidence grow. Even disasters teach you something!
Using HSS tools will teach you about tool grinding (Another excuse to spend your money on a Bench Grinder! )
carbide tools have their uses, but are easily chipped if you are less than careful.
Like others, I am a fan of Tangential Turning Tools. Only one face to grind to sharpen. You can make your own, (There have been at least two designs published in M E W ) or buy the Eccentric Engineering Diamond tool. Costly, but good.
Making tools, such as Tailstock Sliding Die Holders, Centre Height Gauge, Mandrel Handle, etc.can be a useful learning experience, and provide you with useful tools.
When you get a 4 jaw Independent chuck, you will be in the market for at least one DTI and a Magnetic Base. Again, costly, but a capital investment which will be useable for the rest of your life, so buy the best that you can afford., compatible with the amount of use that you will make of them (More than you thought at first! ) .
A 4 jaw chuck will allow you to centre round work more accurately than in a 3 jaw, and to hold square or irregular shaped material..
If your ambition is eventually to make steam engines, it will enable you to make the eccentrics more easily.
But learn the basics, first of all.
An expensive mistake might disillusion you.
Fantastic work has been done on modest machines, some MANY years old.
When it is possible, find a local Model Engineering Club, and join. You will benefit.
HTH
Howard