The draw is pretty self-explanatory the moment you get there. Nisqually Glacier towering above, wildflowers in full bloom surrounding you, tremendous viewpoints, cool air-- there isn't a much better place to be during the summer.
The trailhead to Skyline Trail is located at the north side of the upper parking lot near the Visitor Center and Paradise Inn. You'll see that it breaks off into many other small trails if you want, as well. However, this 5.5 mile hike alongside Nisqually Glacier is classified by the rangers at Mt. Rainier as strenuous. I would say for the first 1/2 mile or so that rings true. It almost feels like a vertical climb at times, but the trail is asphalt for this part so footing isn't an issue. Once you're past this point the trail becomes a lot more tolerable if you're in decent shape.
Your goal should be to get to Panorama Point at 6800ft. From here you get to see anything and everything in the park as well as a good look at the trail loop (also a pit toilet up here). Even a mediocre sunset makes you feel like you don't want to be anywhere else in the world if you're there for wildflower season. I'm not a botanist, so I can only tell you there were a variety of plants and flowers in yellow, green, pink, blue, purple, red, white, beige...it's a spectacular sight. You certainly won't have to look too hard for whistling marmots either. They're everywhere!