Audio Technica AT LP3, Pro-Ject Essential III or , if you want to be able to rip vinyl to a hi res format, a Sony PS HX500
If you're not bothered about hi res transfers, then you can't really got wrong with pro-ject's range really. Audition some in the store with vinyl you know well
Hey finalsound , it's been decades since I bought a turntable so I have no clue what's out there. Back in the day, the best one I owned (and still break out on rare occasions) is a Dual 601. It's a belt drive and was sort of mid market at the time. I've had several cartridges and they all seem to have their own sound. The Shure M91ED (I bought it for $12 on clearance when Shure came out with a new model) was really quite good but my favorite has always been my Stanton 681eee.
I don't know is Stanton is still around or if they still make that cartridge but if you can find one for a decent price, I think you'll be very happy with it.
Cleanliness is of extreme importance with vinyl. Playing a disc that isn't clean is doubly bad. In addition to hearing some horrible pops and click, a diamond hard stylus has a tendency to embed the particles into the soft vinyl.
I always used Discwasher products, but again I don't know if they still exist.
You might want to look for a dirt-cheap turntable and put a nice mat on it for the express purpose of the cleaning operation and keep that separate from the one you use to play the discs.
Also, those who were familiar with vinyl recommended we not play the same grooves again until a moderate amount of time had elapsed. Apparently the walls of the groove are deformed by the stylus but allowing time for the resilient nature of vinyl to return to it's original shape will extent the usable life of the disc.
The preamp can make a big difference. The signal from a magnetic cartridge is rather weak so a preamp with high signal to noise ratio can make a big difference. Also, cheap preamps have a low input impedance that robs you of the sparkly high end. So can cables that have high capacitance. No need to get ridiculous on spending for the cables from turntable to preamp but a few extra bucks can be money well spent.
If you like your music LOUD, vinyl presents a challenge. Acoustic feedback can become an issue even if your turntable is sitting on a stable platform. It's a nightmare if it sits on a cabinet that resonates.
That's all I got. Not much in specifics but maybe I gave you a few things to think about.
Never got rid of it. 
