From Milan take a train in Milano Cadorna to Como Nord Lago then take the C30 bus and get off at Lungo Lago Marconi. Alternatively, take a train from Milano Centrale to Varenna-Essino (about 1 hour), and take a ferry from Varenna's pier (5 minutes walk from the train station) to Bellagio (about 15 minutes, costing 4.9Euros one way). Alternatively take a train to Como and take a ferry from Como to Bellagio.
The C30 bus-stop is near the Como Train Station (Como Nord Lago) on the street named Lungo Lario Trieste. The bus to Bellagio from Como will cost about 3€ and takes just over an hour.
Ferries schedule: http://www.navigazionelaghi.it - there is a fast (40-60 minutes) and a slow (two hours) ferry to Bellagio from Como. When searching for your ferry on the official website you need to check the box in the search parameters to see the fast ferries.
If you are driving, be cautious not to enter the ZTL (Restricted Traffic Zone) in the center of the town, unless you are staying in a hotel in the zone and have prearranged a restricted zone pass. Park before entering the town, as navigating the one way streets without going through the ZTL will be difficult if not impossible.
If you do find yourself driving around a street full of people, be aware that you will also find yourself with a hefty automated fine a few months later, and the rental company will have probably charged you €35 for passing your details along to the police.
Be cautious when choosing your way around and attempting to take shortcuts. Many, if not most uphill paths, end up with the private entrance and no further way, requiring you to walk all the way back. Most notably the nice, wide set of stone stairs uphill right across Hotel Belvedere will NOT get you to Bellagio - it only lead you to the entrances of private villas - after a long climb. Follow the signs to the city center instead.
The Villa Serbelloni is the most popular tourist attraction. Please do not mix it up with the Grandhotel Villa Serbelloni, which has the same name. The Villa Serbelloni can only be visited by a guided tour. You will learn that Napoleon was here and that it was owned by a member of the Rockefeller family.
Take a cuise around the lake.
The most direct way is by ferry (€4.60 each way per person, approximately 10 minutes) to Varenna where you can connect to the Italian rail system at Varenna Esino train station. There are no ticket machines or agents at this station, so you must buy a ticket from the travel agent down the hill from the station url="http://goo.gl/maps/5aL54".