Depending on what happens snow-wise for the rest of the winter and spring, you may not get very high until late in June. The high loop backpacking trips in Grand Teton (Cascade-Death Canyons and Cascade-Paintbrush Canyons) probably won't be melted out until mid-July. There are lots of dayhiking possibilities, though, in the lower canyons.

Check out the websites for Yellowstone and Grand Teton National Parks at nps.gov for trail information. You may also need to make reservations for backcountry campsites. Also google for trip reports and guidebooks for each of these parks. Yellowstone is at a lower altitude than the loop trails in the Tetons, so you'll have a better chance of relatively snow-free trails for backpacking there. Early June may be a bit too early, though.