As a local I love hiking in the alpine of Mt. Roberts. You take the tram up and down. But, there is quite a bit of snow up there this year. There may very well still be snow in June, but the trail should be mostly clear by then. Nugget Falls is flat, easy, and can get quite congested. You can choose East Glacier trail as a less-trafficked but still busy alternative. Mt. Roberts does not require you to take any extra transportation. As an alternative idea you could walk up to Basin Rd and either go up the actual Mt. Roberts trail and ride the tram back down (free with a $10pp spend at the restaurant/bar at the top) or you could continue to the end of Basin Rd to Perseverance Trail, which would be an out and back hike. There were several slides on that trail and the flume trail this winter, but I would think they would be cleaned up by then. Make noise on the trail, in any of these locations.