Vijayawada is famously hot. April through June are punishing — daytime temperatures routinely hit 42–45°C, and the city earns its informal title of one of the hottest places in India. Don't come in summer if you can avoid it.
October to February is the only sensible window. Temperatures sit in the 20s, the air is dry, and the city's main festival — the ten-day Dasara at the Kanaka Durga Temple — falls in October, drawing huge pilgrim crowds to Indrakeeladri Hill.
The southwest monsoon brings sporadic rain from June to September but does not really cool the city down. The northeast monsoon in October–December can produce short downpours.