Budget system around 1K

So $1k for just amp and speakers? You will also need a phono pre most likely to I am guessing, since you mention the Rega.

One possibility might be the Audioengine powered monitors and a preamp...

OK I'll stick my neck out here:
I recommend the B&W 685s used, there is a great review on the TAS website. I own these and really enjoy them. There's a pair on ebay right now...
You need an integrated amp with a phono input. Currently listings on A'gon that I think would mate well with these B&W monitors:
Rotel's ra1062
Creek's 5250
If you can find the NAD c356, it would go well with the 685s also, and it has separate phono stages for MM and MC. Its a little more expensive tho.
Not sure how the Rega gear sounds because I have never heard it. I have only heard the above gear with a CD source, so consider that as well...Happy hunting.

Obviously s/hand will help a lot. Maggie 1.6 or earlier/Celestion sl6se SL600si or sl700si are wonderful. Certain B&W speakers, older Quad floorstanding cone speakers (not the electrostatics simply due to budget), Midrange magic. Totems tighter better bass, and perhaps a bit less magic.

NAD electronics very solid. good luck and pls keep us posted.

I believe the Rega Brio comes with a phono stage and for the $ is quite highly thought of. Epos speakers would also be a good match. Consider budgeting some $ for cables as well.