I think a challenge you may be facing is you're looking for food that isn't characteristically NJ in some cases.

Simply put, you'll be hard pressed to find the caliber of Mexican cuisine that you might readily find in San Diego. It doesn't mean there's not good spots, it's just not as abundant so you may have to drive a bit to find one that really hits all the right notes.

Additionally, while Princeton does draw some level of ethnically diverse restaurants due largely do the university there, it is by no means the best place for any of those cuisine types. You can technically get most of what you're looking for - but I'd hesitate to say it'd be really authentic or really good.

As others have mentioned - New Brunswick has a lot of more authentic Mexican options, Edison is great for Indian and certain Asian cuisines. The Korean food hotspot is really all the way up in Fort Lee, that said, KPot in East Brunswick is not terribly far off from what you can get at your average KBBQ place in Ktown in NYC and a pretty good value. Korean Fried Chicken has become a lot more common than it used to be, if you want the absolute best, you might be heading up to Bergen County, there used to be a spot there called Chicken and Beer that I was a big fan of, but you can likely find something that scratches the itch a lot closer.

Back to my original comment, while there may be a decent spot for BBQ and Soul Food, this isn't the south. Again, you're looking for "good food" that in some cases is not at all NJ's forte. There's been a number of BBQ joints on route 22 in the Union area that have made a good effort, but proven to be unsuccessful. BBQ just isn't as culturally part of life in NJ as elsewhere.

Finally, you can think of NJ as 2 states in one. There's South Jersey which is more in line with Philadelphia and North Jersey which is a bit more diverse and in line with NYC (see Fort Lee example). You're kind of in the middle of mythical "central" Jersey that some claim does not exist (it does).

So yeah, you'll probably have a better time if you head to points north just based on the cuisine types listed.