I gotta jump in here. I don't want to scare you, but to warn you that the weather can be a disappointment in June in the Alps.
We drove the Grossglockner Hochalpenstrasse in mid-June, 2011. We stayed at the highest place you can stay, the Edelweiss Hutte. When we paid our toll at the bottom, the person who took the money said it was just a little rainy on the road.
The weather was miserable when we arrived with sideways wind and rain. We woke up the next morning to several inches of snow, and we were socked in with clouds so we could see nothing. We had to wait to come down off the mountain until someone came up about noon to open the tourist shop and told us it would be okay if we just went slowly. Even the Germans in the serious Toyota 4WD wouldn't try to go down.
It took us quite a few minutes and many turns to get to where it wasn't snowy or icy and was just wet. I was white knuckled the whole drive through the ice and snow, but my husband loves to drive and he had no problem. Even the drive through the snow and ice was beautiful, but I enjoyed the drive much more when I wasn't afraid we'd go off the road. The widely spaced "barrier" rocks would have not been much help if our little rental car headed their direction. We could have driven right between them. Maybe it was just the time of day or maybe they were smarter than us, but I saw no bus of any kind on that road the whole way.
We had also planned to drive over the Stelvio Pass pass in Italy, but I nixed that. We had run into some people in Florence who had to go up twice to be able to drive it and see anything. We did just fine taking the low road on to Lake Como.