Hey there. I did a three person trip in the Bighorns a few years ago. It was my first backpacking trip and actually the first time sleeping outside for me. We had a fantastic time. A lot of people and other trip reports here will suggest the Cloud Peak area, and rightfully so. Just as another option, I'll describe our route. We did Tensleep Lake to Lake Solitude to Lilly Lake and back to near Tensleep lake. Here is a Caltopo map of roughly our route.
Click here to view on CalTopo
We parked at Tensleep and camped just on the west side of Tensleep Creek for the first night. We arrived in the early evening and had about a 40 minute walk or so from the trailhead to where we set up camp. On the second day we got to Lake Solitude, although staying at Mistymoon is also popular. I think from Mistymoon you can make it up to Cloud Peak; I'm not sure what class climbing it all is but there are some folks on youtube who have a pretty comprehensive video of the route.
From Solitude we made it to Lily Lake, which was a great walk and a fantastic spot to camp with great views of Elk Mountain and the surround. From there we headed up to Bald Ridge with the intention of making it back down to our first campsite along Tensleep Creek. On the map I left off the route down because there are several options. We ended up going off trail and just headed down the face of the ridge. You can also take the trail back to Lake Helen and come back down Tensleep Creek, or you can head down the 4WD road on Bald Ridge and catch the trail going down the face of the ridge towards Tensleep Lake.
A lot of people head to Mistymoon Lake and then go through the Florence Pass area. I've never done that route, but there is lots of info on it here and elsewhere.
The trip we took can be done in 3 or more nights depending on what you're looking for. I wouldn't describe the route as difficult; there were some hot spots between Solitude and Lily but nothing crazy. Very little exposure along the whole route except for some very brief spots between Mistymoon and Solitude.
I'd highly recommend the area. Let me know if you have any questions.