You can explore GOA in Nov- Dec, it's a seasonal period there due to new year and all.. and many more activities to explore at that time like speed boat and banana boat rides, paragliding, waterskiing, water scooters, boogie boarding, dinghy sailing, wind surfing, kayaking, scuba diving, snorkeling, and more. There is cruising, and fishing too.
Winter in Sikkim starts from October till march. Thangu, Zero Point, Yumthang Valley, Gurudongmar Lake in North Sikkim are blanked with snow in winters. Gnathang, Old Baba Mandir, Kupup, Memencha Lake, Lachen Village and Chopta valley are some of other place which get snowfall every year in the winter season.
I have visited only South Sikkim in that Changu lake have heavy snowfall. But North Sikkim ki best. Nathu La and Zeropoint in East Sikkim.
Best time to visit- October to March