Can't really add anything new to the wealth of information posted so far, but I can tell you this. I got into machining exactly the way you're about to. I've built several street rods and a few years ago, I needed a custom part milled because I couldn't buy what I wanted. I watched a guy mill it for me and thought this could be interesting. Not too long after, I had my first milling machine and lathe, and it has become my preferred hobby - lot to learn, but I'm having fun.
A word of advice. Within your budget, buy the biggest/heaviest machine your shop can accommodate. My 'go to' lathe is a 13x40, and there have been a few instances where I wished it were bigger. If you're into classic cars, you shouldn't be too intimidated by used equipment, but based on experience, I would suggest being careful to buy a machine that will have easily replaceable parts. It can be a very expensive nightmare if you buy a good piece of equipment that needs what appears to be a relatively minor part but ends up costing a fortune. (I'm currently being very cautious about buying a nice used surface grinder that is no longer in production. If it needs ANYTHING, a good deal could go south in a hurry. Read through my current thread on a K. O. Lee surface grinder, and you'll see what I mean.)
Finally, I would suggest perusing this site regularly. I've learned more about machining from studying the veterans here than I have from any other source. I've never worked in a machine shop, so this is all new. Everybody here is willing to help. Even occasionally, we rookies have something to offer.
Regards,
Terry