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.
Loads of places in india. It all depends on where u stay, how far u wanna travel and what is ur budget. Places like goa, gokarna, pondicherry, andamans etc are gud if yo want a beach experience. For Mountains you can visit off beat places like barog,pangot, palampur, chkrata etc.