My question is, what would I get out of learning functional programming? Is there a good tool or language I can learn that will make me immediatly able to apply my learnings to my daily work in backend/frontend web dev?
If you are interested in Erlang that is a good language to start with. Besides functional programming it also has several good lessons to teach about Distributed systems and fault tolerant programs. Some even call Erlang the only real OO language (maybe besides Smalltalk) so maybe it will make you think a bit differently about what you already know from OO (I know it did for me).
If you were thinking of starting here I can recommend the excellent "Learn some Erlang for great good" which you can read online for free. Google it.
Erlang and Elixir is not the same thing as Elm. Erlang and Elixir are separate languages but are run in the same VM (which is named BEAM) and are therefore usually named together. Elm is a language that compiles to Javascript and is used to create frontend apps for the Web.
Otherwise I have somewhat of a soft spot for clojure myself. It was one of my first foray into functional programming myself and teaching me about func prog at the same time as showing me the elegance of a Lisp will always make it have a special place in my mind. But that is only my personal opinion. If you had been interested in Erlang already you should probably go with that :)