Gangtok is well connected. There are flights to Gangtok from many parts of India, while trains from any part of North India is available to Gangtok. Darjeeling is 4 hours away from Gangtok. The weather in Darjeeling is hot in June. As the heat goes up there are rains. From Darjeeling Kalimpong is 2 hours away. June is a hot month in Kalimpong. As the heat goes up, there is a downpour in June. Meghalaya is called as wettest month in June. You get to see lot of nature spots though it’s not an ideal time to travel in June. Shillong is called as starting monsoon month in June. It is not a cold month, where downpours prevails. Since you are carrying a new born child, kindly check the travel restrictions before booking and travelling. I would advise you to visit Gangtok or Shillong where the weather is not cold but warm though not hot.