In May we took Gianni's Boat tour to see Capri. We took the private 2 hour 15 minute 170 euro gozzo boat tour.
Our captain / guide was Juan Paolo (I believe that was his name!) and he was so present with us, so kind and enthusiastic about showing us all the sights around the island. It was lovely to feel the energy of someone so excited about where they live! He was full of little tidbits and facts that made the tour more interesting. And he was very nice, offering us cold beer and water, and tips on how to relieve seasickness (apparently, by sticking your fingers under your nostrils!).
In May, the water was pretty cold (16 degrees celsius / 60 degrees farenheit) - too fresh for us to swim personally (and there are some jellyfish in the water to be careful of), but there were some brave souls here and there in the water.
After reading some reviews, I was concerned we might have to skip the blue grotto if there was a long line-up, but our captain managed to get us into the Blue Grotto within about 1 minute of waiting - despite the pretty long queue - I don't know how he did it! But we were so grateful he seemed to have connections that helped us skip the queue! Having experienced the Blue Grotto as well as the other grottos around the island, I recommend doing the Blue Grotto (even if it is an expensive 4 minutes!). In my opinion the Blue Grotto is a different experience from the green grotto or the other caves around the island that we visited since it's so dark with only a tiny entrance - it makes the blue light from the water all the more dramatic and special.
I made a video of the whole 2 hour boat tour condensed into 8 minutes. You can find it if you search for "Capri Boat Tour around whole Capri Island, Italy" by JivingGerbil. And if you want to see just a video of our visit to just the Blue Grotto from this tour, search for "Blue Grotto, Capri, Italy | Grotta Azzurra, Italia" by JivingGerbil.
I gave this tour 4 stars rather than 5 only because I felt it was a little expensive (although, I understand this is a fair price for private tours - group tours would be a better deal), but other than that it was a great experience!