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"...

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.
