Oct to February are the month to travel.... but these months will be bit expensive as compare to other months as it's a peak time for Goa... and hotel rates are on peak.
but if you want to travel in budget then go from March to June... because after that till September it's rainy time and most of the shack's and tourist spots are closed.
The best week you can spend in Goa is from 25th December to 1st January. Other than that Goa has a pleasant atmosphere from November to mid-Feb. But still, Goa is a place where you can visit every month of the year and won't regret.