I started my business in the exact same way. I was a little older as I went to the military then college, but bought my first lathe in college, taught myself how to spin up a barrel and now own a company spinning up barrels. There is a learning curve, but for someone with a mechanical and technical mind its really not a steep learning curve at all. There is nothing difficult about spinning up a barrel, its just tedious.
There are a ton of options out there in your price range. I picked up a Romi 13.5 for under 2500$ and man, its one hell of a machine. 50-2500 rpm, 5 hp motor (can take .2" cuts like its nothing), big spindle bore, shortish head stock (can do 21" though the headstock), inch and metric threading without change gears, decently heavy and Romi is a huge Brazilian company with lots of support for the machine as well as extra parts.
Realistically, any good old American machine in good condition will do the trick. Taiwanese lathes like Grizzly or Precision Matthews can produce pretty good darn good results. Nardini is another one I'd look into. Clausing 6908, Rockwell-Delta, Southbend (obviously), Kent and plenty of other options. I'd stay away from Chinese or Japanese built machines. Check auction sights like Bidspotter often. You can usually pick up a good machine for super cheap.
When I was making this purchase, I really wanted something that was fairly simple to work on, had parts available, wasn't going to require constant maintenance, etc. This mostly because I knew next to nothing about machining or lathes in general. So i didn't really want to dig into an older machine. In the long run its really not hard to figure out at all, ask the right questions and practice quite a bit! If you have any questions whatsoever feel free to shoot me a message and I'll help however I can.
As for tooling, be prepared to spend about 1.5K$ for tooling (i have receipts I can send, i think...).
Power converter - unless you have 3 phase, you'll have to find a way to power the lathe which will more than likely be 3 phase. Spend the money up front on rotary converter and don't look back. You'll be glad you did. I am electrically retarded.. like can't wire a light bulb if my life depended on it... and I figured out all the wiring in about 30 minutes.
Where are you located? Maybe I can suggest a smith or two in your area to go see and potentially clean their shop to pick up some info.