LATHES
Try to buy one from and estate sale, rather than a machine shop. The estate sale ones were probably infrequently/lightly used hobby machines that will show less wear than a machine used at a shop day in and day out.
This one was for sale on OK. No idea where you live, but I thought I'd point it out:
https://tulsa.craigslist.org/tls/d/sapulpa-atlas-10-lathe/7158422172.html
It looks like an older machine, but it has a QCGB (quick change gear box), so no change gears. Not a bad thing (my lathe has change gears), it just makes changing the feeds easier. $1200 seems a little high, but that might be what they go for in your area.
This seller looks like they buy out machine shops:
https://tulsa.craigslist.org/tld/d/lenapah-metal-working-equipment-and/7150208764.html
Be wary.
The one is post #14 is worth checking out if you are anywhere near it. It's a good size. A 8-12" swing and a 42-54" length or so is fine for hobby work. For example they are often listed as 9x42 or 12x54. The first number is the swing; largest diameter that can be turned. The second number is the longest piece that can be put between the head stock and tail stock.
Bigger lathes are nice, but they are...bigger. Lots bigger. Which means heavy. That is good; you usually get a nice stiff bed, but they need a really sturdy bench to sit on and are not easy to move. My Atlas Craftsman weighs about 330 lbs. I used 4x4's supporting 2 pieces of plywood 2" thick with a piece of 16Ga steel on top of that to make a table for it to sit on.
MILLS
The plastic gears are often put there as an engineered fail point. Instead of something really expensive and hard to fix breaking, a cheap, plastic gear breaks instead. Lots of people put belt drives on them as well.
Sieg is the company that makes most of the smaller lathes for various manufacturers.
There seem to be a few sizes: the bench top lathes, the Rong-Fu round column ones (and clones), knee mills, then the full-size ones. The first two can be put on a table top (provided it's strong enough), the second two are on the floor. I would look at the first two for hobby work.
SAFETY
You didn't mention if you know anything about these machines, so I would recommend taking a machining class. YouTube is fine, but there is no substitute for learning the right way. These machines are powerful and can hurt you real quick if you don't know what you are doing.
The local community college might have something, even if it's a Maker Space of some sort where you can ask someone to help you use a machine.