What was the first type of government in Cuba?
That depends on how far back you wish to go. Cuba was discovered by Columbus in 1492 and soon became a Spanish colony with a Spanish governor in Havana. In 1762 the British took control of Cuba and held it until it was exchanged for Florida. In any case, since control of the island was held by Spain it would be a monarchy. In 1902 as a result of the Spanish-American War Cuba became an independent republic.