I run my karaoke using software on my laptop. My laptop has an HDMI port, and it can provide an extended desktop experience with that port to one HDMI output – a monitor or a television. Ideally, you want to be able to run the karaoke show tools on the laptop screen and display just the show itself on the secondary monitor. It’s even better if you can display the show on multiple screens simultaneously so that one monitor is turned to the singers who are currently performing and another is facing the audience, so they can see the lyrics too and also so the audience can see the marquee announcing the upcoming singers and other news.
I chose an HDMI splitter I found at Amazon. You want a splitter, and not a switch box for this. The difference is that a switch usually has several inputs but only one output. The splitter has only one input, but several outputs, so you can show the same video on multiple monitors and televisions. This one has four outputs, and was very moderately priced.