The linked post by /u/kieslowskifan will answer most of your questions, but I'll have a quick stab at the last bit. I'll include some references as much of this is paraphrased from an essay I did on this for Uni last year.

Did he try to distance himself from other fascist regimes in order to appear more "moderate"? I hear he is often considered a "semi-fascist", did that play into it as well?

Franco & Fascism is a big topic. Some key points are:

• There were some real Fascists in Francoist Spain, the "Falange". They were the only permitted political party[1] and much like the regimes in Germany and Italy, the party became big.
• However it wasn't quite so active in carrying out large projects (like the "Final Solution") as the Nazi party was, and became more of a social institution that everyone joined in order to get ahead socially & in their careers.[2] Paxton argues that Franco worked to weaken the Falange and subjugate them to his own personal control by making them the sole political party and thus forcing the large conservative, monarchist (importantly both were very Catholic, with the church being much more powerful in Spain than in Italy or Germany)[4] groups to join the Falange and thus change its makeup.[3]
• Payne argues that as early as 1942, Franco was working to remove Fascism from his regime.[5] He had accepted their help in the Civil War (and indeed a large amount of the "first-rank" Falangist leadership were 'lost' in that time)[6] but this can be considered strategic. They had an ideological similarity in that both Franco and the Falange were radically anti-Bolshevik and at least somewhat yearning for an idea of Spain before the Republic[7] (the Republican government could be construed as "anti-clerical", and thus anti-status-quo, and indeed was by Conservative elements)[8], but the original Falange did not like Franco. In 1940, they even tried to assassinate him.[9]
• Franco accepted help in the Civil War from the "Falange" within Spain but also from Hitler and Mussolini. This was not something he reciprocated in any meaningful way.[10] He was opportunistic and ruthless, no doubt, but he was never a committed member of any kind of international Fascist alliance, he was an ally of "convenience" more than anything. He supposedly (I can't find the reference, sorry) offered Hitler that he would join the war in return for some terms that were completely unacceptable to Hitler (large amounts of money, weapons etc).
• Not to labour the point too much, but, Catholicism. There is really no way for Fascism and a strong religious institution to co-exist, and Spain had/has a very strong religious institution. This is the main ingredient that is missing, as Franco absolutely fills many other criteria (Authoritarian strongman, 'ultra'-nationalism, some limited 'palingenesis' A.K.A 'national rebirth', and a 'stab-in-the-back' myth a la the Republican government + Bolshevik partisans towards the Catholic church). Franco was (much like his Portuguese contemporary Salazar) a dictator from the far right of the 'political spectrum', which is certainly similar to Fascism but lacks the fervour, radicalism and indeed expansionism of the very particular ideology of Fascism. Franco was much more "anti-[Liberalism, Bolshevism, Trade-Unionism, and so on]" than he was "pro-" anything (except maybe Catholicism).

I'm not sure I answered your question perfectly well, and I didn't want to produce anything too verbose and inaccessible, so please ask if there's anything you want me to clarify/expand on and I'll do my best.

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[1] Thompson, W., 2011. Fascism. In: Ideologies in the Age of Extremes: Liberalism, Conservatism, Communism, Fascism 1914-1991. London: Pluto Press, pp. 85-103 , Fascism. P102

[2] Payne, S. (1999). Fascism in Spain, 1923-1977. Madison: University of Wisconsin Press. p284-285

[3] Paxton, R. O. (2004). The Anatomy of Fascism. New York: Alfted A. Knopf. P150

[4] Blinkhorn, M. (2003). Fascists and conservatives: the radical right and the establishment in twentieth-century Europe. Abingdon: Routledge. P144

[5] Payne, S. G. (1996). A History of Fascism, 1914-1945. Madison: University of Wisconsin Press. P435

[6] Blinkhorn, Fascists and conservatives. P133

[7] Campos, I. S. (2004). Fascism, Fascistization and Developmentalism in Franco's Dictatorship. Social History, 324-357. P346

[8] Payne, Fascism in Spain. P7

[9] Payne, A History of Fascism. P431

[10] Payne, A History of Fascism. P239

More can always be said, but /u/kieslowskifan has written about this previously here which you may find of interest.

/u/kieslowskifan linked 2 books in their answer;

Payne, Stanley G. Franco and Hitler: Spain, Germany, and World War II New Haven: Yale University Press, 2008.

and

Bowen, Wayne H. Spain During World War II. Columbia: University of Missouri Press, 2006.

​

Which one of these books would this sub recommend?

In addition to the great answers already here, Franco was also keen to leverage Spain into a Cold War partner of the United States. Part of how this succeeded was largely the result of Truman, who hated Franco and the Falange, being term-limited. His successor, Eisenhower, held a more pragmatic view and considered Franco a potential Cold War ally.

Franco, soon after the close of WW2, began making a similar calculation. He carried out some domestic reforms to generate some positive press, and then began a campaign to turn Spain into a premier vacation destination for American tourists. This was heavily aided by much of Europe being decimated by WW2. Franco's government entered into deals with travel agencies, airlines, and hotel chains in order to build up Spain as both modern, open, and a vacation spot. This went a long way into changing the attitudes at least of Americans toward Franco's Spain.

Francoist Spain's external/foreign relations are probably just as important as its internal politics. Franco and the Falange excercised great power quite quickly following the Spanish Civil War, likely due to the sheer exhaustion and toll the war enacted on the country and the brutality of the regime against dissidents. Franco's rule could be characterized as a juggling effort to keep certain blocs of his domestic base both enthused about his governance but also properly bridled and controllable by him. As others have mentioned, Franco began reforming the state over time to curb the power of the Falange, while also offering concessions to them such as prestigious bureaucratic posts, etc. and making certain policy to maintain concord in the ranks. Meanwhile, Franco set about both legitimizing his government, his rule, and his state by marrying it to the Catholic Church and pursuing a "morality agenda" which appealed to Spanish conservatives, the Falange, the military, and cultural revanchist church officials and Spanish bishops. He would dole out a barrage of punishments, concessions, crackdowns, and promotions to each of those support blocs while keeping himself securely entrenched. An early bloc of support of his that he soon kicked to the curb as they posed the most threat to his reign were various camps of monarchists who supported Franco's cause in the civil war and saw him as the means by which to re-establish the crown. Franco often promised them that he'd step down to this end in due course, but eventually disempowered them and had the Falange cannibalize rank and file monarchist supporters. The Falange would later be marginalized in the post-WW2 era to gel with the understandable anti-fascist stance of much of the new world order.

The final part of how Franco's regime become so entrenched until his death 30 years after the fall of fascism was his efforts to economically liberalize Spain and integrate it with the world economy. The Francoist economy is difficult to describe with precision, but initially Franco and the Falange desired to build an alternative model to both capitalism and communism which maximized the benefits of each while minimizing the flaws of each. That model was called corporatism, and relied, organizationally, on individuals building associations based on their industries (its far harder to pin down than this). Mussolini's Italy was the first corporatist state. In light of the emerging liberal capitalist consensus, the growing power of the United States, and a resurrected Europe as a result of the Marshall Plan, Franco coupled his ongoing domestic reforms that permitted greater political and cultural expression with economic liberalization and increased foreign trade. Spain's economy would begin precipitous growth in coming years, and, ironically, Franco relinquishing some control of Spain likely ossified his position in power until his eventual death.

Sources:

Franco Sells Spain to America, Neal M. Rosendorf Fascism in Spain, Seth Payne William J. Callahan, “The Evangelization of Franco’s ‘New Spain,’” Church History, Vol. 56, No. 4