Let $f : X \to Y$ be a morphism of schemes. Is it possible to associate to every closed immersion $i : F \to X$ a closed immersion $f^* i : G \to Y$, such that in the affine case, $(Spec(A) \to Spec(B))^*$ is given taking ideals of $A$ to ideals of $B$ via preimages? Probably this is wellknown to every algebraic geometer :)

It sounds as though what you want is the closure of the image of $F$ under $f$. (That is, the minimal closed subscheme of $Y$ factoring $f$.) If $X =$ Spec $A$, and $Y =$ Spec $B$, and $F =$ Spec $A/I$, and $f$ corresponds to the ring map $f':B\to A$, then we can consider the preimage $J$ of $I$ under $i'$. Consider the set of primes in $B$ containing $J$. Of course, any prime in the image of $F$ under $f$ must contain $J$, since its preimage under the ring map has to contain $I$. Thus, Spec $B/J$ contains the preimage. You can check that it's the biggest such ideal, noting that in order for us to have a map $B/J \to A/I$, $J$ should be contained in the preimage of $I$. Being the closure of the image of $F$ under $f$ is a universal property of sorts (in particular, it's unique), which more or less allows us to argue that this construction generalizes to nonaffine schemes. (Just apply this local construction, and uniqueness tells us that the local constructions glue together.) 

