This is always an interesting topic for me as it is for many of you our age I suspect. One reason is I always explain how my first arcade game experiences weren't in an actual arcade, but rather bowling alleys and pool halls that eventually *became* arcades. Case in point, we had a big entertainment place here in the KC metro area called King Louis West. It had a big bowling alley, a big pool hall, and a big ice rink for free skate, but also figure skating lessons/events and hockey practices/games.
So in the mid 70's 76 or so when my very first memory of that place was when I was 5, the place was pretty much all pool tables with a row or pinball machines. Then you saw a few more pool tables taken out...it seemed like they had over 30 to start...and more pinball. Up until about 77 or 78 then you saw the first "video" games. Atari Stunt Cycle, Seawolf, Breakout..things like this. But the pinball machines still had their place. Then when Space Invaders came out and that was where I first saw it. That was the real beginning of the major change. Once that happened, you had Lunar Lander, Asteroids, and on and on. And it seemed like each week you'd see more and more video games until they filled the video game wall and then you started to see the pinball machines disappear in lieu of more video games. Then you started to see the pool tables disappear in lieu of more video games. It eventually settled into what was more like a dedicated arcade game room that also pinball and pool tables. It still had a dedicated pool area with dozen or more tables, but 1/3 or what it once was. And there was still pinball, but it was maybe one row. If you recall there were still new pinball hits that came out during the late 70's /early 80's arcade craze that were hits across society, not just with pinball players...things like Gorgar, Xenon, etc....so you DID still see new pinball machines show up.
So I'd have to go with King Louis West entertainment center as my first arcade I grew up in.
Now as for the first *dedicated* or purpose built arcade that I visited, I'd have to say it was Nickelodeon at the nearby Oak Park Mall which was pretty new at the time and just few miles away from King Louis West. But the one I probably actually spent the MOST time in was the Fun Factory at the same mall that did in Nickelodeon eventually. I also spent quite bit of time at Showbiz Pizza. We were lucky, it started in KC and the 2nd location in the country was 3 mile bike ride from my house, but it was definitely more short lived than Fun Factory. I also spent time at the Malibu Grand Prix..but much less as it was more expensive. Fun factory opened when I was in 6th grade or something and was there when I left home after high school. I do realize Fun Factory started in other parts of the country earlier as more of an indoor carnival with rides/games etc. But we got it in 81-82 or so and by then it was pretty much a dedicated huge arcade. The nice thing was they were big enough to stock 2 or even 3 of the hottest new games for less waiting.
Here's an artists rendition of King Louis West before it was built(and pretty much what they actually built):
Afterwards they added the "ice chateau".
Entrance was where that covered area to the left was with the spire. To the left (top) was all bowling alley. The biggest area with the raised roof and skylights is the ice rink. The area overlooking it and between in and the skate rink was the restaurant/bar/snack bar and pool room/arcade. The pool hall extended into that square piece on the far right at one point.
LOL..I forgot they called it the "SportsoRama" when it first opened:
Part of the reason I have so many resources on this is that the neat thing is that the building is now the Johnson County KS Historical Museum and Heritage Center for the entire county. The entertainment center was(and building still is) in Overland Park, but it was really just a small suburb city of Kansas City and Johnson County, KS contained a handful of these suburb cities all bordering each other and it really served the whole area. They have cool old signs from actual old businesses on the old drag, photos, memorabilia, for the whole area. (one of the towns is Mission, KS)