There is a bit of a chronology problem in the OP.
Batista was a standard, corrupt, authoritarian disctator with U.S. support.
As the people of Cuba demanded more freedom, some people in the U.S. (including the government) tended to agree.
Castro emerged as the primary leader of the revolution and received a little bit of support from the U.S.
Once Castro was in power, he instituted a number of “reforms” that basically took the country completely into the Marxist camp. (Castro was a true believer in Marxism–much more than, say, Kruschev). This would have irritated the U.S. in and of itself, but Castro then earned himself a lot of bad press by holding public “trials” of the overthrown government and landowners in atheltic stadiums, often followed by summary executions. In going Marxist, he also nationalized a lot of industry that had strong economic ties to U.S. interests.
(He had never hid his Marxist beliefs and the support he had garnered from the U.S. had been the result of policy decisions involving realpolitik and a belief in the U.S. that Batista had to go, but when he actually instituted Marxist policies, he became the bad guy.)
At that point we placed embargoes on Cuban goods.
Since we did not like the “new” (as we claimed) Castro, the U.S. supported a number of Cuban exiles in launching a counter-revolution at the Bay of Pigs. (The understanding was that the show trials should have turned the stomachs of the Cuban citizenry who would be glad to get rid of Castro.)
The actual invasion was bungled; the number of people who actually wanted to get rid of Castro was seriously overestimated; the Cubans perceived the invasion as a joint effort by the (justifiably overthrown) land owners and the (obviously imperialistic) Yankees to overturn all their hard work at getting rid of Batista and Castro’s support was solidified.
Once the U.S. was firmly established as an enemy, Castro openly turned for help to the U.S.S.R. who saw that island as a good offset to the missiles that the U.S. maintained in Turkey, and they moved a number of them in to “protect” Castro from further invasion.
We objected, strenuously, and after some brinkmanship, the Soviets pulled a number of their missiles out of Cuba. (We chose not to acknowledge that some missiles stayed behind so as not to provoke further confrontation and the Soviets did not make an issue of announcing some missiles remained so that they could claim that they had never had that many missiles there to begin with.)
Over the next couple of decades, the U.S. treated Cuba as nothing more than a Soviet colony. (It was certainly a Soviet client state.) During that period, a large number of Cuban exiles established themselves as a political force in Southern Florida and the U.S. has catered to nearly all their whims in terms of “punishing” Castro ever since.