The beach in Varenna is small, but it is right off the Main Street. You get thirsty walk two minutes, have a gelato or a Prosecco and go back to the beach. It is a small town, and this really is for the locals, not because that's the way it's made, but because Varenna is a small, typical town without many tourists. You'll be visiting a true lake town, five minutes by boat from Bellagio.

Other than Varenna for a swim, some of the hotels offer piers and such, but that's not going to be available on your itinerary. The only large public beach on the lake is in Colico. That's all the way at the tip of the Y, and it will rule out Bellagio for you because its a long haul up the lake. It's a big beach, with custard stands, etc, but as such, it's lacking in charm. I would just be happy to have the experience of going on a semi-strenuous hike, grabbing a gelato, then jumping into the lake to cool off. It's doable in a day.

I agree with you, you need to do Bellagio. It's touristy, but beautiful, and is what the lake is known for. As in any tourist town, be careful where you eat. On the outskirts but walkable, are some great places, with spectacular views of the lake, but in the tourist center, buyer beware.

If you are going to eat in Varenna, Cavatappi (corkscrew) is thought to be one of the best restaurants in the region. You'll need reservations for sure, weeks to months in advance, but worth it. But I doubt you'll be able to have dinner on the Lake on a day trip and make it back to Milan, since the good restaurants open late.

By the way, there's a bar along the waterfront in Bellagio that was the inspiration for the Wynn Bellagio hotel in Vegas. But according to the bartender/owner, Wynn has never been there. He did invite the owner to the opening of the Bellagio in Vegas, but the owner declined the trip. Easy to find, and a nice little bit of Bellagio history.