India is too large for a single answer. The north and the south have different monsoon patterns, and the mountains are governed by entirely different rules again.
October to March: the main season
For most of India — and especially for the classic northern routes (Delhi, Agra, Rajasthan, Varanasi) — October to March is the best window. Temperatures are manageable, skies are mostly clear, and the countryside is green from the retreating monsoon. Peak season runs November to February; expect crowds at major sites and book trains in advance.
December and January bring cold nights in the north — Rajasthan nights can drop below 10°C — though days remain warm. The south stays pleasant year-round.
April and May: summer
Brutal in the plains. Delhi in May can hit 45°C. Not impossible, but uncomfortable. The hill stations — Shimla, Manali, Darjeeling, Ooty — come into their own; everyone from Delhi and Mumbai heads there to escape the heat. This is also a good time for Himalayan treks before the monsoon arrives.
June to September: monsoon
The southwest monsoon arrives in Kerala around June, sweeps north, and covers most of the country by July. Flooding, landslides, and suspended communications affect the northeast and mountain regions seriously. Goa in monsoon is grey and wet; the famous beaches turn rough. However, the landscape turns dramatically green, prices drop, and tourists thin out significantly.
The northeast — Meghalaya, Assam, Sikkim — receives some of the heaviest rainfall on earth during monsoon season. Travel there is limited.
Andhra Pradesh and Tamil Nadu receive a second rainy season from October to December via the northeast monsoon. Timing a visit to those states accordingly helps.
Festivals
If you can, plan around a major festival. Diwali (October/November) fills cities with lights and firecrackers. Holi (March) is exuberant and messy and extraordinary in cities like Mathura and Vrindavan. The Pushkar Camel Fair (November) in Rajasthan draws traders and pilgrims from across India. Durga Puja (October) transforms Kolkata into an open-air art gallery and celebration unlike anything else.