Learn Hardcore Turkish: Lütfen termosu getir, çaydanlıktaki çayı içine dökelim. - Please bring the thermos, let’s pour the tea from the teapot into it.

dökmek: to pour (general, especially for liquids).
koymak: to put/place (not used for pouring liquids into a container).
boşaltmak: to empty out/pour out (emphasis on emptying a container).
akıtmak: to let flow/drain (more about flowing).

Lütfen: please (politeness marker)
termosu: termos + -u (accusative “the thermos,” definite object; vowel harmony -> -u after o)
getir: bring (2nd person singular imperative)
çaydanlıktaki: çaydanlık + -ta (locative “in the teapot/kettle”) + -ki (relativizer “the one that is in …”)
çayı: çay + -ı (accusative “the tea,” definite object)
içine: iç (“inside”) + -i (3rd person possessive “its inside”) + -e (dative “into”) with buffer -n- after a possessive: içi + -e → içine
dökelim: dök- (pour) + -elim (hortative/optative “let’s …”; vowel harmony with ö → -elim)

Why are termosu and çayı in the accusative?

Turkish marks definite direct objects with the accusative. Here we mean a specific thermos and the specific tea in the teapot, so:

What does the -ki in çaydanlıktaki do? Why not just çaydanlıkta?

çaydanlıkta means “in the teapot/kettle,” describing location.
çaydanlıktaki turns that location phrase into an adjective: “the one that is in the teapot.” So çaydanlıktaki çay = “the tea that is in the teapot.” This is the standard way to say “the X that is in/on Y.”