Visiting from NY (Mesa, AZ)
A good option for rough kid nights is Guedo's Taco Shop in downtown Chandler. It's not spicy, but they have great tacos and horchata. There's an outdoor seating area with a "cantina" bar, where tired or grouchy kids can run around without making a scene while their parents drown their sorrows. It's not hard-core "authentic" (no tongue or brain tacos here) but with little kids that's probably not what you'd want anyway. Very informal, cash only, and closed Sundays and Mondays.
For spicy food, I second Los Dos Molinos. Ask your server what's hot that day, as dishes can vary a lot depending on the chiles they have available.
JKtCJ had good suggestions; Los Dos Molinos used to be one of my favorites, but the inconsistency and service have kept me away lately. It is definitely *spicy*.
My default recommendation for Mesa fits in well with your request: Blue Adobe Grill. It's New Mexican style, has a nice casual atmostphere, and I've never had anything less than an excellent meal there. Also, they make an excellent Margarita (get yours with Herradura Silver tequila).
Blue Adobe Grille
144 N. Country Club Drive, Mesa, AZ 85201
If you like it spicy, you'll love New Mexico food, something that just isn't found outside of here and, well, New Mexico. It is one of the many subsets of Mexican food, and it's all about spice, especially from Hatch green chiles. If you like it really really spicy there is only one choice, and that is Los Dos Molinos. When they say that an item is spicy, they don't mess around; it is very spicy indeed. They don't have a kids' menu, but they have a quite extensive a la carte menu that would make ordering for the kids easy. You could do a taco with refried beans for about six bucks, or I'm pretty sure the kids would be quite happy splitting a cheese crisp. There is one in Mesa but I'm partial to the one in central Phoenix, which has an atmosphere that can only be described as "over the top". The wait at either one can stretch quite a bit at peak times (and I know with munchkins that can be a serious issue), so go at an off-peak time or make sure you take an extra dose of patience in your bag.
A bit more upscale in the New Mexico vein is Carlsbad Tavern, in south Scottsdale. It's a pretty easy trip from Mesa. They aren't as incendiary as Los Dos, but I think the food is a good bit better there. I don't think of Carlsbad as being as "Mexican" as Los Dos, either. They both have their more Mexican style items, but Carlsbad has a lot more dishes that would be classified more as Southwestern/American than Mexican, while Los Dos has very few. Carlsbad does have a kids' menu, but it's pretty pedestrian.
There is also the possibility of Rancho de Tia Rosa, which is a very pretty place. It hardly qualifies as spicy, but the flavors are pretty darn good. I didn't see a kids' menu on their website, but considering the area they're in they'd HAVE to have a kids' menu.
If you're willing to drive a bit, my money is on Tradiciones in central Phoenix. My folks now look for any excuse to drive 40 miles to central Phoenix from Gold Canyon just so they can have dinner at Tradiciones. Everything about the place is just terrific. The servers are friendly, the atmosphere is quite possibly the best of any Mexican restaurant in Arizona, and the food is as good as the atmosphere as long as you stay away from the combination plates they put on the menu to placate the gringos. In general, the more Spanish there is in the menu description, the better the dish. I don't remember if they have a kids' menu, but as many "familias" as I've seen in there, I'd be surprised if they don't have one. If it turns out they don't, I'm pretty sure that their a la carte selections are very extensive as well.
1602 E Roosevelt St, Phoenix, AZ 85006
3313 N Hayden Rd, Scottsdale, AZ 85251
Los Dos Molinos
8646 S Central Ave, Phoenix, AZ 85042
Los Dos Molinos
260 S Alma School Rd Ste 137, Mesa, AZ 85210