I second the recommendations for Organ Piper Pizza on Hwy 100. Nice family atmosphere and something you don't get to experience every day.
I would steer clear of Stonefire Pizza. As a mom of 3, my opinion of Stonefire is that it's just an expensive version of Chuck E Cheese. Not worth the time...or the money that you spend on everything but food.
AJ Bombers is a winner with my kids and my friends' kids. They love the atmosphere, and it's something different. There are lots of peanuts around, though. So, avoid if anyone has an allergy.
Kopp's is definitely good for the whole family. Save room for custard!
Assuming that you are doing this on a Saturday or Sunday, and that the zoo part of your visit is planned for mid-day, a good place to go with kids might be Organ Grinder Pizza, a few miles south of the zoo on HWY 100.
As an alternative, there is a Kopp's [burgers & custard] location on Bluemound a couple of miles west of the zoo.
I havent been to The Safe House in [literally] decades, so I dont have a clue what it is like today. But when I was in high school and the drinking age had been lowered to 18, The Safe House was one of the easiest places in town to get into underage. It's really hard for me to think of it as any kind of 'destination' just for that reason.
I took my mom to the Safe House last summer; she had been there in the 70s and said it was pretty much exactly the same except they replaced the picture of Burt Reynolds in the Ladies Room (long story and not kid appropriate). As dusty/musty as it was, it was still fascinating although I think many of the spy references your kids won't get. That said, we went for lunch and my burger was really very good. I definitely wouldn't take the kids there after 5 when I think the emphasis shifts from food to drink. Also didn't need the password to get in during the day. Depending on how old your kids are, I think there may be other places they'd enjoy more.
If you're doing this all in one day, I would hands down recommend breakfast at Blues Egg, which is very close to the zoo (about 15 blocks). One of the best breakfast spots in town, perhaps the best if your looking for something above and beyond regular breakfast items. You could also do lunch there. Dinner around the Bradley Center is, quite frankly, a little bit tougher with kids. I've taken my kids to AJ Bombers and had good luck. They were featured on Food Wars, kids liked the airplanes that drop peanuts on the table. I also regularly take them to Buck Bradley's before events, although, admittedly, it's geared towards adults, but it is very large, has a sizeable restaurant area and is quite comfortable with kids.
I don't live in Milwaukee but I recently saw a place on World's Weirdest Restaurants called Safe House.
It's a spy themed restaurant, complete with secret passages and all sorts of crazy things. Looks like fun! Might be good for both adults and kids!
Never been but if I'm ever in Milwaukee I'd like to check it out myself!