I would start with Rega. There are multiple tweeks and upgrades available and many for sale here on AG.
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If you don't mind used equipment, then an used VPI Scout will be well within your price range. For additional savings, you can get an used belt drive TT, such as AR ES-1 or The Turntable, for about $300-500 depends on condition, then you can upgrade with a good arm and MC cartridge with the rest of the money. Happy listening!!
I would second the vote for Rega (a P5) if you want a very solid, nice sounding and simple to use turntable.
On the other hand, if you have a Linn dealer in your area that can educated you and provide support, and you have your eye on upgrading incrementally over time, then I agree an LP-12 is your best bet. A used LP-12 w/ Cirkus bearing upgrade, Valhalla power supply and Ittok tone arm will cost about the same as a new Rega P5 (w/ upgraded power supply) and will sound much better. But there are hazards buying used LP 12's, so without a dealer to educated and support you I would stick with the Rega. The Scout is also a sound suggestion I think, I am just more familiar with Rega.
What is particularly important is finding a dealer you like who will either sell you the gear or support what you buy used. When you are starting out in vinyl, I think good relationships with good people are every bit as important as the gear itself.