Looking to add a turntable to a hand me down system

The Rega will probably be a bit more lively/fast and the Music Hall a bit more smooth/warm. But honestly my approach with turntables is they all sound so good that you're better off focusing on functionality like do you want a dust cover, VTA, how does the tone arm feel, condition/build quality, things like that. Because ultimately you will be able to make it sound pretty much however you want by changing the cartridge.

Then as far as sound goes its usually better with budget gear to err on the side of smooth/warm.

You might consider getting a separate phono amp like the Schiit Mani for $130 (clearer) or U-Turn Pluto (warmer) for $90. It would then plug into an aux input.
They might need a cleaning. SpinClean is well worth it for $80.

Go to Goodwill you never know ,I've seen Pioneer, Sony ,Technics TT cheap....get a new cart and belt if it's not Direct Drive and your good to go....you may need a phono stage ....

Agree with millercarbon. I can attest to his statement, I own a vintage system, Sansui AU 717, Polk Audio SDA's 2B monitors and a Technics SL Q2, but I also have a Music Hall 3.3 LE and a Pro-Ject Debut  Carbon hooked to the Sansui. The tables have a warm sound that compliments the Sui.