In mid-summer, you're going to see quite a few people on most trails. Of course the trails through the geyser basins are the most popular, but definitely worth it. When you're standing there watching a geyser erupt, it's such an amazing site you can block out those around you. Even some of the backcountry thermal features, such as Lone Star Geyser, can have quite a few people on them. Of course if you make the trek back into places like the Shoshone Geyser basin, you'll see fewer people, but that is a very long day hike. (If you hike in, walk the basin and walk back out, you're looking at 20+ miles - that's why you'll see fewer people!) We've found the NE part of the park to be the least crowded. In that area, it's more about wildlife than geological wonders.
If there's one trail I could recommend in Grand Tetons, it would be the one to Amphitheater Lake from the Lupine Meadows trailhead. Ten miles round trip and fairly steep in places, but it gets you an up-close view of the Tetons. Very spectacular.
As noted above, if you really want to get away from the crowds you have to get away from the parks. The nearby wilderness areas of WY and MT are amazing.