Of course claiming some of the lands needs a POD to date back before the Civil War. Cariibean can be done through War of 1812 if the U.S. had Canada so the British Navy could be crippled without their source of wood.
Wouldn't such a timeline be so divergent that a War of 1812 would be unrecognizable from OTL? Besides, if the US had Canada, the British Loyalists would have to go somewhere. To where would they go? The old Carolina Colony? If that was the case, it would even be harder to go that far south.
If the US took the entire of British North America, (again, you'd have to have so many Patriots compared to Loyalists that the PoD would have to be much earlier) the war of 1812 would be drastically different and take place at a different time, if it occurs at all. The loss of Halifax would rid Great Britain of their best port in Canada, and they'd be heavily dependent upon Bermuda.
Granted, that also presumes that Britain cannot find another source of wood, or that its navy wouldn't simply attack the merchant marine where the US couldn't defend.
Anywho, any US gains into the Caribbean, you would have to wait a while. To address Tallest's point about the time of the Civil War, the Ostend Manifesto might be pursued by the Southern states even more than OTL. With Rio Grande being a free state, and most other states from the southern part of the Mexican session being free, save for Yucatan (perhaps), they would feel hemmed in, as he pointed out.
As such, they will try to make it official US policy that the island is to join the United States, quite possibly prompting even earlier and greater responses from the free staters. That could be the conflict that precipitates the Civil War, factoring into Bleeding Kansas (or TTL equivalent. ...might there be an equivalent event in the Rio Grande to try and turn it into a slave state?)
So, before the Civil War, there definitely won't be anything that happens. The US would be having trouble dealing with the Apache and other tribes, on top of integrating a foreign, Spanish-speaking, Catholic population centered around the former RRG, while also dealing with the Mayans in the Yucatan. There wouldn't be the energy for large investments in the Caribbean.
Now, after the war, there is no telling what might happen. If there is a Spanish-American war (Spain could possibly reform a bit in face of the US, considering how much more the US has expanded than OTL. Not likely, but it certainly would be possible) Cuba would be the major focus, as always. They might join if the Lower Southwest feels accepted and treated as equals by the rest of the US. If not, there is no way they would willingly join.
Something that might be interesting is, after decolonization, you have a few of the old British Colonies joining. If the Yucatan becomes a state, it might be interesting if Belize votes to join. I don't have a particular reason for this. I just think the borders would be pretty.
As for other pursuits, what happens to US foreign policy due to gaining that land? How quickly would the US grow. (would there be fewer immigrants than OTL, or more, again due to the Catholic factor) Would splendid isolation continue? How would the other colonial powers treat the US? How would the Central American nations and the South American nations? Would Guatemala take the opportunity to try and regain Los Altos and reform the UPCA? How would Great Britain deal with this US? Would they act the same as OTL, or would they attempt to strengthen Canada as much as they can?
An idle thought in my case, but what would become of the US lease of North Borneo? If the Civil War is earlier (and perhaps shorter, considering the southern free states) would the deal actually be made?