The Spanish Civil War is so fascinating and confusing because of how many factions there were on both sides. Let's go over the three you mentioned:

Falange: explicitly fascists modeled after the Italian fascists. The important thing to note about fascists is that they're not traditionalists, however anti-liberal they are they tend to like modern technology and systems of state organization (which includes economic ideas about "corporatism" which says that instead of struggles between labor and capital they should unite and cooperate). You can find lots of fascist rhetoric about sweeping away the old world can creating a vibrant new one. Part of this was that the Falange was less monarchist than other parts of the Nationalist coalition, while the Carlists were explicitly monarchist but supported a different branch of the Spanish royal family than most Spanish monarchists.

Carlists: while they were very anti-Communist, anti-Liberal, and very very VERY pro-Catholic, the Carlists were not centralists. The core of their strength was in Navarre and while they were not Basque nationalists, during the Carlist Wars of the 19th century they were fighting against liberalism and part of this liberalism (in their eyes) was more modern centralized and bureaucratic government. Although Carlist ideology did evolve over the years, their main ideological pillar was traditionalism, so they wanted to turn the clock back (way way back) in a way that the Falange didn't want. A part of this was that while Carlists were strongly anti-democratic they preferred more local and traditional forms of authority than the fascist Falange.

Francoism: the important thing to note about Franco is that he wasn't originally the head of the Nationalists. Sanjurjo was originally supposed to be the leader or at least a figurehead but he died in a plane crash right off the bat and even after that there were important generals like Mola and de Llano who didn't want to take orders (at least initially) from Franco. The Carlists and the Falange also organized their own militias. Franco became more prominent as his Army of Africa was much more successful than the other Nationalists forces, but to control the overall nationalist side he had to manage a pretty broad coalition between the army, fascists, traditionalists, more normal conservatives, etc. Francoism wasn't a pre-existing ideology but rather something thrown together ad hoc to try to hold the Nationalists together. Francoism later developed during Franco's dictatorship but it wasn't something that existed at all before the war.

As far as Franco being a fascist that's hard to nail down as fascism is notoriously slippery ideologically. A lot of this is because fascism can be pretty incoherent or self-contradictory on a lot of issues so it's more of a style and a set of tactics than a coherent set of ideas. While you can certainly make an argument that Franco was fascist and many historians have and he certainly allied with the Falanage, used a lot of its rhetoric, and turned its leader Primo de Rivera into a martyr after his death he didn't really take them seriously and tried to keep them away from the main levels of power. Just being an anti-democratic authoritarian by itself isn't enough to make someone into a fascist which is why a lot of historians see him as more similar to the various non-fascist military dictators you can find in Spain, the Spanish-speaking world in general, and the world at large.