Start Here
https://www.awgb.co.uk/clubs/ find your local club get along see what is available in your area.
Second hand lathes exist if you have a van to pick them up, as do sharpening systems and tools. Some time at your club will hopefully get you a hands on and a better idea of where you want to jump in, and maybe even a link to local second hand kit to get you started.
I would hate to sort out how much I have spent on Woodturning over the last decade, but it has been money well spent mostly thanks to the local AWGB and some local turners that let me play with their toys before I started collecting my own.
But as a starter for you...
Any lathe will do, but size weight, motor power will depend on what you want to make, I would recommend that you get a variable speed version but that has been pretty common for over 20 years now so lots to choose from.
Some form of jig based sharpening system will be handy especially in the early days, but a cheap 6" grinder will do with practice.
You will need a Roughing Gouge, a Bowl Gouge, a Spindle Gouge a Skew (19mm oval is my main choice) and a parting tool. Expensive ones are great but as you will be learning to sharpen them the first set wont last as long as the second set!
You will need safety glasses and when sanding a dust mask, you can start with the paper ones but...
You will need abrasive and polish, it is important that some of your early work looks great, this helps the addiction take hold.
Once addicted the cost will be met, tools will accumulate and time evaporates.
Over time you will find yourself the owner of a chain saw, bandsaw, extraction system, a drawer full of finishing products, possibly an air brush, pillar drill, and.... but you will be happy