October and early November are the best months to visit Delhi. The monsoon has cleared, temperatures are in the mid-20s Celsius, and the air is (relatively) clean. This window lasts about six weeks before winter fog begins and air quality deteriorates sharply. March is also good — spring temperatures and reduced smog — though it shortens quickly as April heat arrives.
Avoid Delhi in May and June if you can. Temperatures regularly hit 45°C and the city feels hostile. The monsoon (late June through September) brings some relief from the heat but adds humidity and disrupts travel. November through January brings cold nights, heavy fog, and the worst air quality in the world — Delhi's AQI routinely reaches hazardous levels from October burning season through January. This doesn't stop people visiting, but plan outdoor sightseeing for the middle of the day when smog lifts slightly, and bring a good mask.
Independence Day (August 15) sees the Prime Minister speak from the Red Fort — the surrounding streets close and access to Old Delhi is restricted a day ahead. Republic Day (January 26) brings a major military parade down Kartavya Path (the renamed Rajpath); this is spectacular but requires tickets well in advance and comes during the worst smog months.