Ok, technically not IN Goa but only one and a half hours by train from Margao and the most amazing and hospitable village. Recommended by every career hippy I met, Gokarna was described to me as the place to go if you want to see what Palolem was like in the sixties. It didn't disappoint. In particular, the picturesque palm fringed Kootlee beach (get a boat from the main beach or walk a steep-ish hill) is a must - even in high season you often get the whole place to yourself. If you can get there at the end of February for the Hindu festival it will be even better.
Get the train from Margao or Chaudi (Canacona on timetables). Travel 3rd class for the friendliest and most entertaining journey.
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Lots to do from here as a base camp, Karnatika and Jog falls. Gambling on the ship in Panjim, stay at the Panjim Inn, wonderful Portugese building. Great restaurants.
Mandrem up north is still ok or even better Dui in Gujarat, fly from Bombay on jetairways 30 quid.