Rajasthan: it is considered that Rajasthan can be visited all year around. Monsoon months are July and August, but rains are short downpours so they probably won’t spoil your vacations a lot. It could be slightly cold from December till February especially at the evenings and early mornings,so bring warm clothes with you. It gets much warmer in March, while in April it is really hot.
Sikkim: best months to visit this very small but picturesque state are from March to May and from October to November when the temperature is comfortable and not so cold as in winters. Monsoons period is from June to September.
Tamil Nadu: this is probably one of the hottest state in India with a temperature raises up to 40 degrees from April till June. The best months to visit Tamil Nadu are from June till September and then from January till March when the temperature is not so hot and there are no rains. Rains come in October and last till December.
Uttarakhand: As in the other regions, there is a monsoon period in Uttarakhand with rainfalls that lasts here from June to September. The best time for traveling is from March to May when the weather is warm enough with a temperature around 30 at daytime and 15 at night. Winters here are cold with temperature maximum of 15 degrees and lasts from October till February.
Uttar Pradesh: Try to avoid to travel to Uttar Pradesh during monsoons period from June to September. as monsoons period could turn into real disaster here because the rivers overflow their banks. Months from October to March are considered to be the best to visit Uttar Pradesh, but don’t forget about warm clothes - it can be cold in the evenings. April and May could be very hot, but this is a perfect time if you are planning to visit mountains.
West Bengal: Best months for visiting west bengal are from october to march when the temperature is not so hot. Keep in mind that in the evening temperature could fall down to 10 degrees.
If you are going to visit India for the first time you will probably find interesting and useful my article India travel tips.
For the information about transport in India and the best way to move around the country, please read my article Trains and buses in India.