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"...
 A: Is a function that takes an element from V and gives you an element from W.
A: The phrasing $T:V\to W$ in plain language means "$T$ is a manipulation of things that, when applied to something in $V$, gives something in $W$". When we speak about this manipulation, called a function, we say "$V$ under $T$ goes to $W$".
The arrow symbol $\to$ in the phrasing represents the idea of ".. goes to ..".
A: It means that T is a function(or transformation), mapping elements of V(Domain) to elements of W(Codomain).
A: 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.    
A: Suppose we have T : V $\rightarrow$ W 
Well this means that it is like a function between two vector spaces that takes element in V and sends that to an element to W it could have more properties as every element gets hit that is T(V) = W that means your transformation/function is surjective. 
Now if we have also that the surjection is minimal that is it is injective then we have an isomorphism which means that both spaces are the same only difference between them is the way we write the letters in each vector space.
For example suppose we have the following linear transformation from $R^2$ to $P_1$ given by 
T : $R^2$ $\rightarrow P_1$ given by T(a,b) = a + bx
This gives an isomorphism since first they have same dimension so all we have to check is that it is injective and I will leave that as an excerise for you as it will be good to improve your intuition.
