Stop for supplies, Walmart (Pocatello, Blackfoot, Idaho Falls), super Target (exit 319 in salt lake) on the way.
You may consider going all the way to West Yellowstone, only two hours further than Idaho Falls.
The first day you will want to do the Geyser basins including Old Faithful, no fishing the first day, waters here are too hot this time of year. Idaho Falls-West Yellowstone-Madison Junction-Firehole Lake Drive-Old Faithful-Madison Junction-West Yellowstone.
If coming from Idaho Falls, get and early start. You'll be ending the day back in West Yellowstone. Two hours to West Yellowstone. You can stop, breakfast, check in with fly shops or go right on in. Plan is to head to Old Faithful and work the way back doing most of the stops.
7/20 morning. Go to fly shops first thing and get fishing permit, flies information and advice. This day will end at Roosevelt Lodge. West Yellowstone- Madison Junction-Norris Geyser basin-Mammoth (terrace and village)-Roosevelt Lodge. Optional stops Artist Paint Pots, Virginia Cascades Drive, Sheepeater Cliffs, Roosevelt Arch, Gardner, Wraith Falls, Blacktail Deer road, Petrified Tree.
7/21,7/22 One day Lamar Valley the other Canyon and Hayden Valley.
Lamar Day (good day for fishing) Wildlife watching, many locations, Trout Lake short hike, big hard to catch fish, chance to see otters. Optional Cooke City. Fishing Soda Butte creek, Lamar River Slough Creek. Optional hikes Yellowstone River picnic and Specimen Ridge or Hellroaring Creek.
Canyon Day: Roosevelt-Tower Falls-Dunraven Pass-Canyon-Hayden Valley-Canyon-Dunraven Pass-Tower Falls-Roosevelt. Optional choice of two hikes, Mount Washburn or Cascade Lake. Cascade lake good fishing (Grayling).
7/23 Roosevelt-Canyon-Fishing Bridge-Lake Hotel-Bridge Bay-West Thumb-South Entrance-Kelly, WY. Optional side trips. Lake Butte, Storm Point, Natural Bridge, Elephant Butte, Lake scenic cruise.
7/24-- someone else can take it from here.
Fishing is artificial lures/flies. Some waters are fly fishing only. All native fishes are catch and release only. West Yellowstone is home of many good fly shops. Gardnier has Parks fly shop. Fishing supplies at the general stores inside the park are limited to say the least. When you are in the park avoid the Firehole and Madison inside the park, the summer heat and thermal waters make the water too warm for fishing. Much better other places. Good spots: Gibbon River (Gibbon Meadows, Elk Park, Norris Picnic area, above Virginia Cascades), Obsidian Creek, Lava Creek, Lamar River, Soda Butte Creek, Slough Creek, Cascade Lake.