Help…looking to buy turntable

I purchased from Underwood their LSA 2 that is made by Edwards Audio. Its [essentially] the guts of a Rega with an improved tone arm and acrylic platter.

https://www.underwoodhifi.com/products/lsa-turntables

I’m assuming your $1200 budget includes cartridge.  I would consider it wise to get the best possible TT, and then get the best possible cartridge without going overboard.  In other words, don’t put a souped-up engine in a Ford Pinto.  
The suggestions for the Music Hall Classic Seem very sound (no pun intended) to me.  It’s also a very visually appealing TT, if that matters to you.

You could install (and would have to learn how to optimally calibrate/align) say, an Ortofon Bronze cartridge on that bad boy, and, with tax, be around $1100.  You could opt for the Ortofon Blue and be around $900.

I’m only suggesting Ortofon cartridges because they seem like good bang-for-the-buck cartridges. On an old TT, I had a low-end 2m Red which, back then, retailed for about $100.  Surprisingly good cartridge.

Other posters here, I wouldn’t doubt, may have better suggestions for cartridges in, say, the $200-$500 price range.

Rega +1

Tonearm pre-installed and suggested carts make them easy. Oh yeah, great sound too. 

VPI Cliffwood or Player - entry level VPI tables, both with cartridge pre-installed. The Player lists at $2000 (over your budget, but does also have a built in phono preamp and headphone jack).  The Cliffwood is $1500, but you'll most likely need a phono preamp...Musical Fidelity V90-lps at around $300 is a really good entry level phono stage that supports both MM & MC.

I'd look at a Technics sl1200C ($1200 with a decent Ortofon 2M Red cart).  It has a built-in phono pre, but for an additional $129 you could get a Schitt Mani and that should keep you going for a while.
The tonearm has a SME style head shell connector that makes cart changes easy and fun.  The 2M red is a good starting place (I have a Hana SL on my 1200 GR, a remarkable cart for the price, but I also have a Mofi MasterTracker that sounds different but just as good, and several Grado and Shure carts that are fun to rotate in).  There are many carts between $200 – $800 that would work well with this table!