The Cuban Revolution of the 1950s was two things at once. Firstly, it was a rebellion against the illegitimate government of Fulgencio Batista (and the puppets he used to feign legitimacy) after he initiated a coup on the 10th of March, 1952, which overthrew the constitutional order and even 'temporarily' depose the constitution of 1940 itself. Secondly, it was an attempt to remedy long term problems within the Republic.

Regarding the first issue, Batista's coup and dictatorial government (1952-1958) occurred immediately before elections which would likely have swept reformist politicians into power. Obviously this irritated members of the "authentic" and "orthodox" parties, who stood to win the most from the upcoming election. The following years would be ones of economic prosperity but also brutal repression of free speech and violence against all opposition to the regime.

Regarding the second, more long term, problems in Cuba, the republic had decades worth of issues with corruption and economic inequality, dating all the way back to the first years of independence. The overthrow of dictator Gerardo Machado in the 1930s and the creation of the very progressive (for its time at least) constitution of 1940 seemed to set the country on the right track. But by the end of the 1940s the country was once again mired in corruption and the only half resolved issues of inequality seemed near the boiling point. The Orthodox Party (of which Fidel Castro was a candidate) seemed poised to battle corruption, and potentially enact some economic reforms, when the coup ended their shot at gaining their shot of controlling congress and maybe even the presidency.

Fidel and most of those who attacked Moncada (1953) were members of the Orthodox Party or students, all mostly middle class, with only a few working class people joining in. The attack itself failed militarily but the violent repression used against survivors of Fidel's group turned popular opinion against the government. Of the dozens captured, many were tortured and outright murdered in custody as an example. Famously, Haydee Santamaria was shown her brother's eye and boyfriend's testicle as a part of her torture (both had been part of the attack force).

By the time Fidel surrendered, public opinion had swung even further to his side (thanks to Batista's vile acts against those he had captured) and enough pressure and scrutiny was on the regime that no further murders or outright torture were done against new members of the group that attacked Moncada who were captured or decided to surrender.

Fidel then turned the trial against him into a political circus with a political speech which eventually became the pamphlet History will Absolve me.

Though pacific protests against the regime continued until 1958, a significant sector of the population favored armed resistance. Some groups fought in the cities, like the Directorio Revolutionario (armed wing of the student union; think the equivalent to the Weather Underground in the US). Others, like Fidel's 26-6-M, fought in the mountains (Fidel and others were freed by an amnesty c. 1955 and quickly returned with a new force and arms in 1956). This separation between rural and urban guerrilla warfare is often referred to as the difference between el llano (the "plain", meaning cities) vs la sierra (the "mountain range", meaning the countryside).

While urban guerrilla fighters fought bravely they also suffered heavy losses and very limited successes. Famously, the leaders of the Directorio were shot to death by Batista's police and intelligence forces at the apartment building at Humboldt street, number 7. It has been immortalized by a photo of a child looking at the blood trickling down the stairs from the murders. This, it should be said, Batista reacting to the almost successful attempt on his life by a group of Directorio members who attacked the Presidential Palace late in the Revolution.

The movement that had actual momentum, however, was rural guerrilla warfare. Unable to face Batista directly, they were basically able to make big towns isolated militarized islands with little control over the countryside. They had a lot of popular support, especially because they espoused rural reforms that (due to lessening government control over the countryside) they were even able to apply in certain areas.

As time went on and the Batista regime showed itself to be corrupt, bloody, etc., dissent spread slowly but surely throughout the army. It began to disintegrate, refuse to follow orders, or even see individuals join the rebellion. It should be noted that dissension in the army was nothing new, such as when old guard attempted a rebellion in 1956 against Batista, known as the "Conspiracy of the Pure Ones" (Conspiración de los Puros) or just as the failed coup of 1956.

In any case, increasing popular discontent in the cities, the loss of even minimal control of much of the countryside, the dissolution and insubordination of the army, and mounting American pressure for Batista to leave and be replaced by someone less hated (but still pro-US of course), all left Batista in an increasingly unsustainable situation. Then forces allied to the 26-7-M movement, under the command of Ernesto "Che" Guevara, blew up the armored train that was delivering supplies and troops to the Eastern war effort and succeeded in capturing the city itself. It wasn't just a matter of losing control of the countryside: He was losing cities too. The war was lost.

Batista quickly fled on New Years eve, December 1958. He first tried to flee to the US but Washington was still angry that his refusal to step down had let potentially radically revolutionaries take power in Havana. Instead he fled to Trujillo's Dominican Republic. Only later would he resettle in Miami.

Fidel, leading the strongest rebel faction at the time, was sure to rush troops to Havana and Santiago de Cuba in order to make sure it was clear who won the day. The next few years were spent consolidating his control of the new government and performing purges (in the sense of expulsions and jail time) of those to his left and to his right (purging old Communists, often Stalinists, who didn't tow the line when he remade the Communist Party in his image and eliminated the leaders of the nationalist but liberal democratic side of the 26-7-M).

Source-wise, Lars Schoultz' That Infernal Cuban Republic gives a good overview of this period and the American role in pressuring Batista to step down.

Richard Gott's treatment of the period in his Cuba: A New History, isn't half bad as an introduction for non-academics.

I would strongly suggest the article in the Cambridge History of Latin America that deals with Cuba during the Revolution.

If you can read in Spanish, I would try the experiment of comparing Sergio Guerra's Cuba: A History (available on Amazon) with Rafael Rojas' Historia Mínima de la Revolución Cubana. The former is very pro-Revolution while the latter is very anti-Castro, however both books are from recognized and respected Cuban academics (though only overviews).

Hope it helps.