Before you spend money on a lathe, you need to decide the use to which it will be put.
If your ultimate aim is to build a 6" scale traction engine, a big lathe will be needed, but it won't be too much use if you want to be a watchmaker, or vice versa!
Some make Locomotives, some Traction Engines, others, Tools, or unusual mechanisms.. A great many restore Motorcylces, Cars or Stationary engines in 12" to foot scale.
Each to their own!
But once you have experienced the joy of owning a lathe, and the world of opportunities that it opens up for you; you will, never want to be without one!
Your budget needs to allow for tooling, not just cutting tools, but measuring instruments, such as micrometers or callipers, and Dial Test Indicators and a stand (probably a Magnetic base ) Then you will start to hanker after Taps and Dies!.
Once you get a 4 jaw independent chuck, DTIs become a necessity for really accurate work.
Bear in mind that a 3 jaw chuck is unlikely to hold work absolutely concentric.
Also, for the sort of money that hobbyists are likely to spend, you will not be getting an heavy duty industrial, high precision machine. A £500 machine will not be the equal in precision, quality or longevity of one costing £20,000..
Having said that, with care, fantastic work can be done on quite modest, and elderly machines.
You can learn a lot of the basics (and make mistakes with minimal cost impact ) by making a few basic tools, such as a Centre Height Gauge, Tailstock Sliding Die Holder, Mandrel Handle, Tap Wrenches, Die Holders.
Better to scrap an embryo Tap Wrench that an expensive casting!
They will all be useful, and in making them you will learn, and gain confidence
Don't forget a reasonable Bench Grinder for grinding things like High Speed Steed lathe tools, and when you become proficient, to sharpen your drills.
Hope all this rambling helps
Howard