Harriman State Park is one of the most spectacular and overlooked state parks in Idaho. It's located along the famous Henry's Fork branch of the Snake River and has an amazing bounty of plant and animal life that is often passed over due to the park's proximity to Yellowstone National Park. Even though it's not as well known as Yellowstone, I'd still recommend I visit to Harriman State Park if you want to experience some truly wild plant and animal life.
Harriman State Park is famous among locals for three main reasons: fishing, Mesa Falls, and cross-country skiing. The Snake River is famous throughout Idaho for its abundance of rainbow trout and if you're looking to strap on your waders and do some fly fishing in the great outdoors, then Harriman is the place to be. Also, don't miss a morning at Mesa Falls, a spectacular and massive waterfalls about an hour away from the park entrance. As you arrive, there are several viewing platforms to enjoy different angles of the falls. Finally, cross-country skiing. I myself have never personally gone cross-country skiing but I know that Harriman is a favorite among those that have.
Finally, there are horse outfitters near the park and a day riding along the trails is a great way to spot moose, elk, and the dozens and dozens of swans that use this park as a nesting ground.