Get yourself a nice old technics with an S-shaped arm.
Any technics will do. Buy an Audio Technica AT95E cartridge and put it under the technics.
The AT-PL120 won't do any better, believe me
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If you want to buy something new (for whatever reason), spend that little extra money and go for the Technics SL-1200. If you don't have that money, look for a Pro-ject or a Goldring turntable. Again, fit them with the AT95E cartridge. It's bang for the buck
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Please do note: You have to have a phono connection on your amplifier or you'll have to buy a phono preamplifier.
The sound that comes from a record player is a lot softer than the sound that comes from a CD player. That's why you need the pre-amplifier.
As a pre-amplifier I would suggest the NAD PP1, 2 or 3. There's cheaper ones, but this is the cheapest one I can think of that doesn't sound like utter crap.
You could try the Pro-ject Phonobox. I haven't heard it so far, but it is said to be quite good.