Hi Paul,
I hesitated in replying as I am a relative novice compared to those who have commented already but I would like to second the advice you've gotten from Les, Twisted Trees and Bill21. My first lathe was a cheapie and due to its poor build quality, vibration in the headstock etc. etc. etc. I never managed to really produce anything that I could say I was proud of or happy with. This was despite having taken a two-day course with a professional tutor.
I took the plunge and bought a decent lathe and found that - with hard work, practice and perseverance - I am now producing pieces that I am much happier with. I am now only limited by my own ability and skill, rather than by the lathe. And I know that, whilst I have a long way to go in terms of developing my techniques and skills, I am likely to reach my own limitations before I reach those of the lathe.
In my opinion, if you have the desire and the means, I would recommend picking up a decent second-hand lathe.
Woodturning is not a cheap hobby to get into - the costs of a lathe, chuck, PPE, turning tools, a sharpening system and dust collection all add up. I personally would recommend buying the best lathe you can within that budget. Once the initial investment has been made you then only need a source of suitable wood, which can be as cheap or as expensive as you wish.
Sorry for rambling on, I got a bit carried away there lol
but I hope this might be food for thought,
Regards,
Mike.