This can't be understated. If you've ever watched the Giro much, you'll see that it can be awfully cold at higher elevations, snowing on the peaks, etc. That said, the Giro doesn't seem to be a "national holiday" as the Vuelta does, so it's likely to be a bit less crowded. Personally, I'd go with a company like Thompson Bike Tours that specializes in doing tours around big races, especially if it was my first time. Once you get a lay of the land, it's much easier to plan things like this yourself.