"I have a 7 and 9 year old who are both reasonably active and can hike up to 5 miles without issue."
In the Park starting at the String Lake Trailhead hiking Hidden Falls comes in under 5 miles. Add Inspiration Point if you can. Around String Lake is another one. Adding Laurel Lake still comes in around 5 miles but that's a rugged climb the Park Service doesn't like to be recommended. Taggart and Bradley Lakes is also in that wheelhouse. "Around the Block" from Grand Teton Climbers Ranch wonderful. The Ranch is a great place if weather becomes a drag at all. One day going SE of Jackson to Granite Creek in the Gros Ventre Mts. is AWESOME. Great hikes there. Spectacular Southern Crest of the Granite Hi-Line is 5 total miles with views that will knock your sox off. First 2-3 miles of Swift Creek also special. Hiking East side of Granite Creek from Swift Creek Trailhead also quite special. Awesome Boondocking down there too.I hope to do the Hi-Line Monday or Tuesday. Early Season Favorite Soak in Hot Springs afterwards. I can share a photo album privately. Hi-Line favorite early hike up higher.
"They both like fishing, but my 7yo loves to fish."
Streams are pretty blown out right now. Shore fishing below JAckson Lake Dam and North of Colter Bay great. Friend did very well at the inlet from String into Jenny Lake the other day. If you rent a canoe String with the SHORT and EASY portage to Leigh Lake is incredible day. No motors on Leigh, truly magnificent.
"We plan to bring our bikes so we would be interested in good biking routes."
Great Pathways system, check friends of pathways. If you bring Mt. Bikes love the bumpy old "road" down to the Historic Bar B C Ranch and RKO road. Also old two tracks from Science School RD to Mormon Row, Kelly Warm Springs two track to Mormon Row. Probably too early to hike Mt. Leidy but biking those back roads above The Hatchet is incredible. Do NOT forget your bearpsray there or anywhere.
"We will plan to visit Yellowstone since it so close, anything we absolutely should not miss?"
Most folks focus on Old Faithful, DO NOT miss the Brink of the Lower Falls at The Canyon. If its a day trip head out morning civil twilight and be prepared to be out until evening civil twilight. Do your best to avoid driving in the dark of this greatest ammalian habitat of the Temperate Zone in AND out of the parks.
"Are mosquitoes likely to be out of control?"
Don't forget your cancer in a can, DEET.
WHERE ARE YOU GONNA CAMP?
) planned this year. First camp will be near Slide L, which is towards the higher end of encounter probability. The rest of the time we'll be above tree line, so I'll leave the spray behind. If I'm solo during hunting season, I pack it. Too much chance of a carcass.