I too went down the Karaoke road a couple of times. The first was a CD based player with some cheap mics. We did it with two other couples who are very good friends. After a few refreshments, to loosen up the vocal cord of course, we all had a good time.
Sometime later the group decided the CD based system was limiting song selection and spontaneity; and one lady in particular (who sang weekly in a Choir which did performances half a dozen times a year to audiences of up to about 1,000) wanted better quality sound for her voice. As I am the resident geek of the group, I was selected (pushed) to figure out how to make this system happen. As each couple was to host Karaoke Dinners, the system has to be portable, and able to hook into various home stereo systems. (We shared the base costs amongst the 3 couples. So there was "budget" pressure to get it done but keep costs down. I hated that as I like a bit of a free hand with system design.)
My solution was very similar to AcuDefTechGuy but with a Behringer 12 Xenyx 1202FX Mixer. (About $155 US at the time.) As there could be a few refreshments involved I decide to purchase a set of 4 wireless Mics that had decent reviews. (We couldn't have folks tripping all over those cables, and twisting them up handing off Mics to each other. Safety First!) I found using the "Low Cut" filter was critical in preventing feedback loops when Singers turned their "Open Mic" towards the speakers.
I used an older Laptop of mine as the nerve center with DJ/KJ Software, and on board & USB connected libraries of tunes. I typically used the Mixer's USB Output => PC to get the Microphone sound into the Laptop and used it to blend with the Tunes before outputting via a cable splitter into video for the TV, and analogue (RCA Cables) to the "Home Stereo". With PC based tunes, selection was quick but after a few refreshments, the "crowd" often couldn't figure out what they wanted up next, so I ended up being the KJ (what a term) and sort of a
Name Sing that Tune evening evolved. I never got to printing a list of tunes of my Karaoke Library. (My Bad) Of course setting up and operating the gear was my job, as it was well beyond the others (especially after a few refreshments!
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After only one cycle of evenings and a second one at our place (about 4 in total -months apart as I remember), it was getting near Christmas time and other things became top of mind, and the gear - now mostly forgotten - sits in storage in my "Man Cave" - as my wife calls it. (Junk Room is what it's become, as half of it is still full of clutter with stuff "temporarily" stored there from our major Reno a couple years ago...) The Lady who wanted better sound could never quantify want she wanted added / subtracted to her voice; so I think that dampened her enthusiasm. I suggested she come over to our place and I'd setup the gear and we could tweak the settings and hopefully come up with a sound she liked. That would become a base line EQ just for her. But that never came to pass and a couple of times since, she's suggest we look at more dedicated gear for Karaoke to eliminate the Home Stereos, as she somehow thinks the cheap Karaoke/PA gear she's seen at a local Pub would sound better. But I don't think the rest of the crowd concurs.
In hindsight, we should have spent more time on optimizing the EQ settings. I typically try for "Flat" but you can do amazing things with one of these mixers. It's certainly not my main HT system that's the problem (especially now that I've upgraded it all since the Karaoke days), and I think the dedicated Karaoke Gear purchased was just fine.
Maybe I should dig it out and try setting it up again with a newer Laptop. At our house I could just run it all from a Laptop into my Denon AVRX-4400H and let it split it into video and audio as I have always feeding the TV in that room. Simpler is usually better.