Most of the victims were shopping for Eid-al-Fitr, which marks the end of the Muslim fasting month of Ramadan, Nepal Das, Tripura's inspector general of police, said.
Hindus were also out in large numbers, stocking up for the annual religious festival of Durga Puja which begins on Saturday, he added.
Television footage showed men carrying bleeding victims including women and children to hospitals.
"At least two of the blasts were powerful," Das said, adding that the serial explosions were the first such attacks in Tripura.
Das said an investigation was under way and a massive hunt for the bombers had been launched in Agartala.
"These were bombs and we are trying to ascertain what kind of explosives had been used," he said, adding that the blasts all occurred within 45 minutes.
The latest explosions came after a wave of attacks in Indian cities, the latest of which was a bomb blast that killed seven people and wounded 82 in western India last Monday.