I think they are most often called karaoke bars, but it very much depends as there is a wide range of establishments of different types that offer karaoke. It might be a bar, a pub, a lounge, etc. It might (as is often the case in the UK) a "karaoke night" (go to a karaoke night), which is where a pub or club puts on karaoke for just one night, perhaps weekly or monthly, etc.
The choice will be determined by the circumstances, therefore. I'm not sure that "going to karaoke" will do, put it is possible, I suppose. Perhaps others will have opinions on that.
I would say "I'm going to a karaoke" + whatever the place might be -- as the mole listed.
Note that karaoke machines are readily available for people holding karaoke parties at home as well.
"I'm going to karaoke" (note lower case, it's not a proper noun in English) might mean something specific in Japan, but such is not the case in the U.S. where it would be inadequate without further information)