Verona too Varenna

Check
www.Rome2rio.com for starters

Looks like there is a bus from Bellagio

I am not seeing any options by train without a change in Milan. The best I could see was Verona to Varenna through Milan with one stop. The bus would take 2 hours longer. The stop you want for Varenna is Varenna-Esino. If you are then leaving from Varenna by train, get your onward train ticket when you purchase the ticket to Varenna as there was no train ticket office as I recall in Varenna. You may be able to buy a ticket on board the train but I can’t be certain of that. Edited to add there is a ticket office in Varenna but it may or may not be open. If you don’t have a ticket, find the conductor before he finds you to buy a ticket or you will get fined.

There is a route from Verona to Varenna by train with changes at Bergamo and Lecco if you are determined to avoid Milano Centrale. It takes about 10 minutes longer than the Milan connection.

But what is your purpose in avoiding Milan?

There is a travel agency selling train tickets inside Varenna-Esino station, open Monday through Friday. The local tobacconist may sell local trains tickets.

I bet that even in Varenna they all buy train tickets on trenitalia.com these days. An hotel with free wifi would be a good idea.