Are spaces with isomorphic fundamental groups homotopically equivalent? I know that the converse of this statement is true but I am not sure how to go about finding out the answer to this question.
 A: Assuming you're taking a course at the moment in algebraic topology, you will soon be (or have already been) introduced to the homology of a space which is another (family of) homotopy invariant of topological spaces, in the same spirit of the fundamental groups. With this invariant, you will find that the first homology groups of the one-point space $\{0\}$ and the $2$-sphere $S^2$ give $H_2(\{0\})=0$ and $H_2(S^2)\cong \mathbb{Z}$, which implies that $S^2$ and the single point $\{0\}$ are not homotopy equivalent, even though they have isomorphic (both trivial) fundamental groups.
This is a recurring theme. Although the fundamental group can distinguish many spaces up to homotopy, it is not a total invariant. It's just one of many that have been defined and studied by those working in algebraic topology and abstract homotopy theory. You can think of the fundamental group as a good first step for trying to distinguish two spaces, but should be used in conjunction with the many other, more powerful invariants of the subject. The real power of the fundamental group, in many cases, is that it's readily computable and often easy to visualise.
A: Compare $\mathbb{S}^2$ and $\mathbb{R}^2$.
