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What would this process of gathering the meaning of a sentence be called? What would the segments derived from the sentence be called?
"John and Derrek both love cake"
-> John loves cake
-> Derrek loves cake
"John was mad that the weather was rainy today"
-> John was mad
-> weather was rainy today
It seems like what you have in mind is components of the sentence. (In semantics, "implications" refers to a broader range of things: all the propositions that must be true if a given sentence is true.)
The phenomenon you're describing is known as "compositionality": that the syntax and the meaning of the parts combine to yield the meaning of the whole.
Since that process is called "composition", we might call the reverse process "decomposition".
Note that in modern syntax and semantics, there's no guarantee that all the elements that appear to transparently combine actually do. The underlying structures might not match the surface ones.
John and Derek love cake.
⇒ John loves cake.
⇒ Derek loves cake.John and Derek met.
⇏ * John met.
⇏ * Derek met.
What is it called to derive all the implied meanings from a sentence?
Your examples do not derive all implied meanings of the sentences (e.g. John loves cake implies the existence of John, cake, a certain binary relation, etc., all separately)
It seems that what you have in mind is called reducing a semantically complex sentence into elementary ones that constitute it. The qualification semantically is important, as "John and Derrek both love cake" is, differently from the other one, not a complex sentence syntactically. I'm pretty sure there's no single word for this kind of reduction, other than maybe in some obscure branch of formal semantics
