To the OP, I'd like to reinforce that YES, you can enjoy vinyl sound very adequately with a cheap, used, vintage TT and a new cart.
I personally own a Dual 1237 from the 70's and just recently purchased a Grado Prestige Black cartridge for it. TT cost me 20 bucks and some elbow grease to clean it up from sitting in someone's basement for 20 years. Cartridge cost me $60 from my local hi-fi shop. The TT came with the original Audio Technica cartridge still mounted, and while it got me by for a month or so while I researched new carts, when I put the Grado on, it sounded like I had a new TT. The sound was 100% better. Like everyone else is saying, it may not be "audiophile" quality, but it will be MORE than awesome to get you started, and far, far better than the sound you would get from a brand new TT at that budget.
Vinyl Engine | The Home of the Turntable will likely have a pdf manual for you to download for whatever table you end up finding.
Search flea markets, garage sales, resale shops, donation shops, etc. for good brands. Sony, Pioneer, Denon, Onkyo, Dual, Technics are all made great turntables in the 70's and 80's and tons of people in North America owned them, so they should be fairly easy to find. With a little bit of looking around you should be able to find something that works.
Also, keep in mind that if you do not have a phono input on your amp/receiver, you'll need to purchase an outboard phono amp. My NAD C340, does NOT have one, so I bought a this:
Needle Doctor 1-800-229-0644, Esoteric Sounds Rek-O-Kut Professional Phono Preamp MKII
Again, likely not an "audiophile" quality phono amp, but more than serviceable for my budget setup.
Lastly, and I can't stress this enough, look into and spend some time researching different methods to clean your vinyl records. You're likely going to be buying a lot of used vinyl, and the cleanliness of those records will have the largest impact on the sound quality. Whether you buy a $500 record cleaning machine, or read up on the many ways you can clean them for free in your bathtub, do it. You'll really appreciate it, and it'll help build that relationship to your music and enhance that "vinyl experience" that all of us vinyl junkies really love.
Good luck.