Burano is an astoundingly beautiful place. Not to be missed. (Murano is not quite as nice as Burano, but if you have the time, a visit there might be fun also.)
Museo del Merletto (Burano Lace Museum):
We went to the islands of Torcello and Burano in May 2016. Torcello was lovely...quiet and serene that allowed time to contemplate what Venice was like before "Venice". Burano was instead very touristy with a lot of ticky-tacky stalls one after another, surrounded by cafes packed with tourists. The fact that you are traveling in winter may be a saving grace! There is one beautiful and genuine Venetian lace shop, EMELIA, on Burano...highly recommend stopping there.
The only economical way is by vaporetto (water bus). Line 12 runs from Fondamente Nove.
A Tourist Travel Card ACTV will cover the trip.
Consider buying a Tourist Travel Card ACTV, which entitles you to unlimited use of the waterbuses (vaporetti) serving Venice and the islands of the lagoon, and to unlimited use of ACTV Mestre land bus and tram service to and from Venice and in Mestre itself—and land buses on Lido. (A trip on a vaporetto without the card would cost you €7.50.) The card does not cover train travel.
ACTV Tourist Travel Cards, which can be purchased either on line, at the airport, in Mestre, or in Venice, are a tremendous bargain that will pay for themselves many times over during your visit to Venice. The link below will provide information about cost and card selection, regardless of where you plan to make your purchase. (If you purchase online you must print and later exchange a voucher for the actual Tourist Travel Card at designated ACTV locations.)
The Rolling Venice discount card is for young people between 6 and 29 years of age. (Children below the age of six travel free.)
The validity of ACTV Travel Cards starts from the exact moment of the first validation.
You must validate your ticket before you get on the vaporetto. If you do not validate the ticket you can be fined.
To validate the ticket, pass the card in front of the sensor in the center of the validation machine before entering the boarding area; a green light and a beep will confirm that the validation has been successful. A red light and buzz indicates that there is a problem and the ticket has not been correctly validated or has expired.
And you must also pass the card in front of the sensor in the center of the validation machine before entering the boarding area for every subsequent trip.
We visited Burano on a rainy Nov day last year when piazza San Marco was a mini pool but Burano was not flooded.
It's a pretty little island. Visited early in the morning for least crowds.
The painted house lined canals look much prettier when it's not overwhelmed with people.
You can see photos of my time in Burano on yogictravel.wordpress.com if interested.
Have a great trip!