Happy to help. I grew up in the park at Mammoth and went to jr/sr high school in Gardiner. As for places to eat... Gardiner's not known for culinary excellence, but here are a few options: 1) Yellowstone Mine is the nicer restaurant in town. Located at the Best Western hotel. 2) Ironhorse Cafe has a patio that overlooks the river and serves sandwiches, salads, etc. 3) There a few pizza places. Outlaw's Pizza is across from the Yellowstone Mine. There's a new pizza place on Park street which has a patio and faces the Roosevelt Arch. And then there's the K-Bar which is a bar/pizza joint that sits right on the main drag. 4) There's also a BBQ place that's pretty good. It's a log building that rests just a block from the K-Bar adjacent to the "High Bridge" over the river.
Hikes - This is much more fun question to answer. If you include the area around Mammoth in the park, I could fill your entire trip.

Some of my favorites are: 1) Bunsen Peak, which starts up on Swan Lake Flats just south of Mammoth. It's a short hike (probably 4 miles roundtrip), but provides a lot of bang for your buck. You get a great look at the park and A-B Wilderness from the top. On clear days you can see clear to the Tetons. 2) Across the road from the Bunsen Trailhead is a hike to Soldier's Corral. This is a very pretty walk that takes you up over a sage-covered ridge and drops you over toward the headwaters of the Gardner River. It's a really pretty place to take a family hike and bring a sack lunch. Beautiful views of Electric Peak. Grizzlies tend to like that lush little valley, though, so keep your eye out. 3) If you guys are into seeing some wildlife without being surrounded by clueless tourists, then I'd recommend hiking up McMinn Bench and walking the crest of Mount Everts for as long as you desire. Start at the Rescue Creek Trailhead near the North Entrance to Yellowstone (trailhead is only like a half-mile from the North Gate). Cross the footbridge over the Gardner River and then leave the trail and just start climbing the big slope on the north face of Mt Everts. You can see bighorn sheep up there. I've also seen antelope, elk, black bear, red fox and bison on various hikes in that neck of the woods. You won't see anyone else when you leave the trail. You can take the crest of Everts clear back up to the Lava Creek Picnic Area east of Mammoth if you're up for a good walk and you have someone to pick you up. 5) If want to do a little fishing on your hike and you have impatient youngsters who will want to join you if the action is non-stop, then I'd head to Knox & Fish Lakes. These lakes are north of the park in the A-B Wilderness. You can get to the trailhead by driving up to Jardine and then following the signs for "Bear Creek". The road dead-ends at the trailhead and both lakes sit in gorgeous spots at the base of Castle & Monitor Peaks. Tons of little brookies to catch. 6) The best hike in terms of scenery, in my opinion, would be the hike up to Pine Creek Lake. The trailhead begins at the Pine Creek Campground which is closer to Livingston than Gardiner. The scenery is unreal, though. Beautiful views of the Absaroka Range and a double waterfall at the top near the lake. Just awesome Montana stuff. The hike is STEEP and about 10 miles roundtrip, but it's worth it. Just beautiful.
As for other stuff to do, I'd probably take them on an afternoon float down the Yellowstone. These are fun and you can do longer trips that go through Yankee Jim Canyon which provides some whitewater excitement. There are a number of rafting outfits in town. We use the Flying Pig. Also, you might want to check out Chico Hot Springs. It's halfway between Livingston & Gardiner near the small town of Emigrant. It's actually close to the Pine Creek Trailhead so maybe you combine a soothing dip in the pool after your big climb up to Pine Creek Lake.