What does $T:V\to W$ mean in vector spaces? What does the sign $\to $ mean in contexts like: "show $T:V\to W$ is an isomorphism" or "if $T:V\to W$ is a linear transformation"...

First and foremost $ \to $ is the symbol for a map. In our case the map $$T: V \to W$$ is a linear map, i.e. $\forall \; x,y \in V, \; \lambda \in K,$ where $K$ is some field, we have $$T(x+ \lambda y)=Tx+\lambda Ty.$$ For simplicity we are taking the vector spaces $V$ and $W$ to be over the same field $K$. The field can be thought of as $\mathbb{R}$, or $\mathbb{C},$ for concreteness. The linear operator $T$ is defined to act on the whole space $V$. 
Now, we want to further classify our linear map. Does $T$ preserve distinctness, namely is $T$ injective (one-to-one)? Does $T$ map onto the whole space $W$, namely is $T$ surjective (onto)? If $T$ is both injective and surjective, we say that $T$ is bijective. 
Suppose that $T$ is bijective. Then, the inverse operator $T^{-1}$ is well defined. Composition of $T$ with $T^{-1}$, results in the identity operator. If these conditions on $T$ are met, then we say that $T$ is an isomorphism of vector spaces.
Remark: I called $T$ a linear operator for pedagogy. In practice people often just say operator, where the linearity of the operator is implicitly assumed. Another phrase for $T$ is linear transformation as you mentioned in your question, or linear function.    
