Slough Creek is a place of superlatives. It's jaw droppingly beautiful, chock full of large native Yellowstone cutts, and -- if you take the time and effort to hike to the famous Meadows sections -- just remote enough to make you think you've somehow slipped into a time machine set to a day in pre-Columbian Wyoming.
You'll fish Slough generally from about July 4th through, oh, maybe mid September. While you can fish downstream of Slough Creek campground, if you're in reasonable physical condition you'll want to take the hike to the First, Second, or Third Meadows. So you'll need a daypack. In it, pack a rain jacket, your largest Nalgene water bottle (or a smaller one and a water filtration device), a bag of beef jerky, sunscreen, your fly vest or chest pack, a reel -- you get the idea. I like to also pack my Patagonia goretex hippers and wading boots -- though it's possible you'll never need to enter the stream.
On your person, you'll need bear spray as this is prime grizzly territory. It's part of the deal. Far more dangerous, however, are moose -- which you'll likely encounter on the hike up.
Park at the trailhead, which you'll see on your right before you get to the campground (look for the sign in box), and make a fairly strenous one hour hike to the First Meadows. A less strenous extra hour will get you to the Second Meadows, and another easy hour of walking will get you to the Third Meadows. Plan to get to your destination by about 9:30am as the fishing will generally start about 10am, so back up your start time accordingly.
Regarding flies, while you can have a good day with general attractor patterns, I'd check with a local fly shop before heading in. Blue Ribbon Flies in West Yellowstone is a great option and you can even call and have your flies shipped to you if are not planning on taking the long drive to West before going to Slough. Their number is (406) 646-7642. An online option is a really cool outfit called Fatties on the Fly: https://fattiesonthefly.com/. My buddy Ben will be more than happy to set you up if you choose that route.
One last thing: On your hike in, you will occasionally see a horse-drawn wagon with a potentially cranky driver at the reins. They do not give rides, so don't ask.
While often overlooked in favor of places like the Madison from National Park Meadows downstream to Seven Mile Bridge, the Baker's Hole and the Cable Car runs, the Yellowstone at Buffalo Ford, Muleshoe Bend on the Firehole, the Lamar Valley and Slough Creek, the Gallatin near its confluence with Specimen Creek is a must-fish spot. A cold river only in peak form for a few weeks per year, the Gallatin often fishes well at exactly those times in the summer when the Firehole and the Madison shut down due to overheating. Some bushwhacking required to get to the spot where Specimen Creek enters the main river. Highly recommended.