From what I understand the forces that fought the Aztec and besieged Tenochtitlan were mostly composed of rebellious vassals of the Aztec, with only a small amount of Spanish soldiers. Once the Aztec were defeated, how did the Spanish go about defeating the rest of the Mexican tribes?
Good question. There's more room for discussion, but I've written on this quite a bit before, you might find these answers of interest (heads up, they are long-ish):
On main forms of resistance after the fall of the Mexica/Aztecs, including the Mixton War
On forms of Spanish political control, and continuing indigenous influence. See also some of the follow ups that relate to your question.
As a short intro to those: I'd highlight that I) while there was continuing military resistance especially and northern regions and Yucatán, central Mexico was under relatively firm Spanish control by the later 16th c. Still, this allowed for continuing Aztec political (and cultural) influence. And that II) the Spanish never held control over all of modern-day Mexico, with the mentioned resistance, but also frequent riots on central cities sources of fear for the Spaniards. So it's not simply a question of one group defeating the other, but rather of complex new societies emerging from violence and social/cultural intermixing. These two points I talk about more in above answers.
Hope this helps! In case of follow ups let me know, I could get to them tomorrow (getting late here).