GV64, where do you live? My wife's fam is in Portland and we're still debating traveling there the weekend of Ok game. Long shot, but would be fun to meet up and have mini-watch party if it worked out. Our kids are 13 & 15 (older girl, younger boy). We live in Philly now and the kids have only been to 1 game each in Neyland, been to Knoxville a few times separately. We're trying take them to the homecoming game this year. All the other suggestions are great, but since we have kids with similar ages, here are the things mine like and most of these are pile-ons from others.
- Thursday afternoon or esp Friday by midday, Market Square is fun. Typically some decent buzz by Friday as out of towners are showing up for football game weekend. Lots of shops and restaurants. Lots of orange crap to buy.
-- There's a store called "nothing too fancy" that has some great UT, Knoxville, Smoky Mountains, etc. swag. Hats, shirts, etc. and they're all kind of unique, funny, cool, etc. Not officially sanctioned nike stuff at all. For example, you won't find any player jerseys here, but you might find a shirt that has a Tennessee pun on it or some dolly parton t-shirts. We always end up spending a lot of money here.

-- Nothing super special but Cruze Farm ice cream shop is an easy walk from Market square. My kids (well my wife and I too!) all love it. It's a local dairy farm ice cream shop. You can blend flavors like a frozen yogurt shop, but it's not self-serve and it's infinitely better than frozen yogurt. You can still sample stuff though despite it not being self serve. Always a line at night and when it gets busy, which tells you it's good. The owners are married couple and wife is a UT alum. Don't fact check me.
- Calhouns on the river gets a bad rap from locals. It's kind of overblown, food isn't anything amazing, I completely agree. But for an out of towner on a football game weekend, I think it's almost a must. Don't go before midday on Friday or you wont get the buzz. After noonish and beyond on Friday, you'll start seeing the Vol Navy traffic pick up. Especially if it's a nice sunny day like it's supposed to be, go there and only sit outside on the patio, deck. If you're not hungry bc of the awkard time you're there (after lunch but before dinner), we've had a drink date with our kids. Mom and Dad had a beer (or 3!) and kids got sodas. They loved watching the boats and most of the time some of the boaters will be up on the deck eating or drinking and usually are happy to talk to the kids about where they rode the boat in from, etc.
- Bookstore is also a must. Overwhelming amount of merch. Warning: whatever you think something might cost, multiply it by 1.75 and now you're probably close to the actual price. Expensive. Even if you don't buy a ton though, it's fun to walk around and see all the stuff. You can spend hours in the on campus bookstore. Plus it's fun to people watch. Lots of football fans from out of town like yall will be there, and then you'll see random students walking around there in their PJs and messy hair even though it's 3pm in the afternoon.
- Gameday: Get there early enough to see what you want to see, but pace yourself. It's not the most massive of games, so you can prob sleep in a bit if you can. Campus will still be buzzing early, but with the 415 start, the game likely won't be over until after 8, so esp with kids I personally wouldn't aim for an arrival much before 11/12 unless everyone is cool with a hot sunny exhausting, but fun day. Your kids are older but it's like spending all day at Disney...if you can keep up and make an 18 hour day enjoyable, go for it, but if everyone (including the parents) are going to meltdown by the evening, then it's worth it to me to slow roll the mornin . Also depends on where you're staying, parking, etc. Can take hours to exit the campus if you park in a garage, depending on where it is.
-- Volwalk. My son is into it. Daughter isn't. I know the players get hyped seeing kids there, and for many it's a must see. Can get crowded so have to get there a bit early, but even at your kids' ages, folks are generally reasonably accommodating at letting kids sneak through towards the front as long as they don't look like adults yet.
-- Watching the pride march in. I think it's a must see. Route changed recently so make sure you have the gouge on the new route.
-- Fan fest/Vol Village. My kids always love this. Tons of stuff to do, games to play, lots of free swag vendors give out, contests, etc. Usually there is live music and they'll be broadcasting other games on a giant screen somewhere. When it's just my wife and I we often skip this, but if you have a half-hour to hour you need to kill anytime before walking in the stadium, this is always a great place to go. Right at the humanities bldg. Basically between the library and circle park, you'll hear it before you can see it.
-- I'd def start heading to your gate ~ 1 hr prior to kickoff. Inside of that, it's hit or miss on the waits and can be a total zoo chaotic disaster getting in. The earlier you get in, the more likely you can head down the first rows and watch the players stretch and warm up, etc. Can get some great pics. But then you miss out on lots of the fun stuff outside the stadium. The longer you hang out outside, more fun you have out there but then the busier and more stressful getting into the gates will get. So pick your poison. Honestly if you're in your seats 10-15 mins prior to kickoff, you'll still get to see most of the band do it's thing, etc. but we've learned better to be early heading to the gates than late.
Pro-tip: Hodges library has the best public bathrooms close to the action on gameday if you're near it!
Obviously this is all just my own opinion. All the other food and activity recs that others have posted are spot on too...Have a blast!
Edited to add, I remember now my son has been to 2 games, daughter only once in Neyland, but i think they've been to the stadium on gameday maybe 3 times each now? A few massive games where they hung on campus for the buzz, then went back to a friend's house while mom and dad went in to the game bc tix were too much $$! Thus the things my kids have liked/disliked despite not actually attending a ton of games. Yet!