Firstly, do not buy - as in DO NOT BUY - one of those cheap tinplate lathes with twin square bars for a bed. Badged as Clarke, Sip, Draper, Lumberjack, Aldi and probably others, they are not fit for purpose.
Go online and look for a lumberjack lathe at around £140 and you will see the type I mean. I have read so many bad reviews from owners of this model.
Look for a decent brand such as Record, Jet, or one of the Axminster range ( preferably the trade range if you can afford it), and go down the second hand route if you can't afford new - far better than being tempted by a nice, shiny, new piece of cr@p.
In terms of second hand, there are the makes I mentioned above, plus Myford ( ML 8 or Mystro), Tyme, Arundel and Coronet. If you have the pennies then a union graduate or a denford viceroy will last your lifetime. All these are discontinued now so be aware that accessories might be getting scarce. Check that you are getting all the necessary parts ( faceplates, banjo, tailstock etc).
From a personal viewpoint, I started with a Black and Decker drill attachment and it was AWFUL. Then I managed to get a Myford ML8 which was wonderful by comparison, although I suspect anything would be wonderful by comparison. That was over 30 years ago and I still have it. I also have acquired along the way a union graduate and a viceroy, which is why I mentioned them.
Secondly, visit the truly excellent website run by Tony at lathes.co and you will get information on practically every decent wood lathe ever made ( and metal lathes, milling machines etc etc).
You will find details on spindle threads to know whether currently available accessories will fit - Tyme and Arundel might be problematic here.
If you buy a second hand lathe without faceplates, and later decide you need them, then expect to pay £25 - £45 depending on make, model and size. My viceroy has an outboard thread of 1 1/4" x 9 tpi and I don't have a faceplate so I bought taps to make my own out of wood - cost me £40 but means I can make as many as I need.
Thirdly - if you see a decent second hand lathe going complete with chuck, turning tools etc then don't underestimate the value of those extras. A chuck could easily be worth upwards of say £70, and new turning tools from £20 to £60 each, possibly more. ( Again, DO NOT be tempted to buy a set of 6 or 8 new turning tools for £25 - they belong with the lumberjack lathe ( i.e. in the dustbin).
Fourthly, you must have some means of sharpening your turning tools. Turning with blunt tools is tedious and unproductive, and will turn (?) your hobby into a chore when it should be enjoyable - and you will think it is your fault for not being skilled enough when in fact it is the tools that are the issue.
A bench grinder with a decent rest might be enough, certainly the cheapest route. You might need to get better wheels and rest though. Then there are sharpening kits that go up to hundreds of pounds, for example the Sorby Proedge at around £350. I don't have one but have read loads of reviews praising them. Whatever you decide, you simply must have some sharpening kit. Don't expect to take your tools to the weekly club meeting to be sharpened. You might need to touch up an edge after only a few minutes turning on dry, hard wood.
Hope this essay is helpful. Definitely check out that website. Definitely avoid that cheap kit. And finally
Enjoy turning.
K
Post script
I jst read the original post again and noticed you are mainly interested in bowl turning. I would not recommend the ML8 for this, I am not a fan of the outboard turning set up which is why I bought the graduate.
You might want to consider a lathe with a swivel head. I have never used one, but it would mean you only use the inboard side so no need to buy accessories for both inboard and outboard ( usually not interchangeable). I believe some Record, Jet and Axminster models offer this facility. The older models I mentioned do not.
K