Hi @gumby05 - Thanks for reaching out!
In my opinion, there aren'y many places around Shoshone that won't be impressive! It's such an incredible area.
For your 5-7 day timeline, the Gannett Peak trail is worth taking a look at. This is a technical trek, so if it looks like one you and your friend would like to do, make sure to do your research and be prepared before heading out. Contacting a park ranger beforehand about trail conditions, any important updates, etc. is always a good idea. They would be a good additional resource to tell you all of the trail options that fit your criteria.
If you're keen on sharing, we'd love to hear about how the trip goes once you are finished this summer, no matter where you end up!