You've gotten some good advice so far. I especially like BJ's suggestions. I would add that, since you are in the vicinity, you visit Lake area; it's quieter. If you want a cook's night out that won't totally break the bank, have a meal at Lake Lodge cafeteria; the food is quite good and - for Yellowstone - not too pricey. Nice view from the rustic dining room too.
You're in a perfect spot for a nice, short trail: Storm Point. It takes off from the park road a few miles east of Fishing Bridge. DO ask about bear conditions before you go, as this trail is near Pelican Creek, which is grizzly habitat. I'd advise buying some bear spray (expensive but effective) at the General Store there at Fishing Bridge. If you are hiking/walking for exercise, remember to pace yourself; the park is at high altitude, most of it between 7000-8000 feet elevation. Talk to the folks at the FB campground and who work in the store; also to those working at the Yellowstone Association store at FB - they'll all have good suggestions for you & your family. Fishing requires a license for those over a certain age (15?) - make sure to inquire. There are short-term licenses that are inexpensive.
Another good way to get some fishing in, though a little more expensive, is to go out on Yellowstone Lake with a fishing guide. Check the Xanterra website, or the Yellowstone one at npr.gov, for more specific information. You can keep all the lake trout you want.
There is no "rush hour" from outside the park on the east side, as most people staying outside the park come in from the west. As previously mentioned, traffic is slow throughout much of the park and will be heavy in some locations. Animal jams are frequent and make great photo ops (but stay in your car if the animals are right on or alongside the road). Spending time in the evening in a pulloff in Hayden Valley, watching animals or even just the sunset, is a delight; you're in the perfect spot for that too, and I recommend it.
Plan on driving lots of miles and having plenty of scenery, wildlife and other tourists to watch.
🙂 You have lots of walking opportunities in the various thermal areas - Old Faithful is just the tip of the iceberg - with all the boardwalks in those areas. Make sure the kids STAY ON the boardwalks!!!!!!!!! Can't emphasize this enough. You'll get a park newspaper when you enter the park - READ IT. It prescribes the distance you need to keep from the various animals, plus lots of other helpful and important information. Park visitors who end up in trouble invariably did not read & heed what's in there.
I suggest spending one full day visiting the Old Faithful area, walking around the geyser area (as much as your party wants). If only some want to walk, there are plenty of places for the others to wait and relax. (Relatively) inexpensive food at the General Store or Snow Lodge's "grill," and the ice cream at the General Store can't be beat. West Thumb is worth at least an hour or two, Canyon at least 1/2 day, Mammoth and Norris at least several hours each. Lamar Valley, the northeast section of the park, is a prime wildlife area and where you are most likely to spot a wolf (never guaranteed - I've been to the park umpteen times and still haven't). Look for a group of people off the side of the road with spotting scopes on tripods.
🙂 If you're on a stricter budget, as mentioned there are lots of picnic areas around. And for the elderly and similarly challenged, lots of toilets (pit, primarily, but NPS keeps them pretty clean), both in picnic areas and by themselves.
Have fun and enjoy! Look at the traffic slowdowns as good opportunities to enjoy the scenery, get a good long look at buffalo (often close-up), etc. Five days will be enough for you to hit the highlights, especially since you plan a couple days in the Tetons in addition.
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