I know that if two spaces are path-connected and homotopy equivalent then their fundamental groups are isomorphic Does this go the other way so we can say that if two spaces have isomoprhic fundamental groups then they are homotopy equivalent?
No. Consider two spheres $S^n$ and $S^m$ with $n,m>1$. Then $\pi_1(S^n)=\pi_1(S^m)=0$ while $S^n$ is not homotopy equivalent to $S^m$ whenever $n\neq m$ (which can be seen by simple calculation of homology groups).
For more sophisticated example consider the double comb space $X$, which has all homotopy groups trivial $\pi_n(X)=0$, homology groups trivial $H_n(X)=0$ and cohomology groups trivial $H^n(X)=0$, but it is not contractible (it is not homotopy equivalent to a point).