The variety of comments is pretty indicative of the depth of knowledge and experience of each poster.
Charles Lamont is absolutely correct in his practice and recommendations. One bought a lathe which had the basics of a a faceplate, a catchplate and a pair of centres- hard and soft. If it was a screwcutting lathe, the cogs arrived as well.
There was none of this three jaw 'blether' and the next accessory was the 4 jaw independent chuck and the first discovery was that it was a damned sight easier to use two keys - and not the supplied one. There was none of this silly excuse for a fancy tool post ' gazira gazinta' and a simple one tool fixing went on the tool post. One stuck a used single safety razor blade between the work and lathe tool tip- and got that right before romping off into outer space with the pissing match gadgets.
Despite the possible comments, that was a wonderful learning platform on which the rest of one's hobby was based.
I still have my home made collets, the book by Sparey had the drawings and a note of 'when you have the basics- and not until' you could move to making a 5cc diesel engine- to his design. There is a serious remark- out of old age and experience. A three jaw chuck is only accurate when new of holding perhaps 5 thous concentricity. If you stick a piece of rough round metal in to it, you are straining and making so called accuracy worse. Again, if work is held by the tips of the jaws you knacker the holding accuracy further. The end result is a wailing letter to the favourite forum of 'how do you correct your balls up?' The answer, despite all the wonderful words, is that you cannot.
End of lesson
Norman