Suppose that $M$ is a closed connected 3-manifold and there is a degree 1 map $f: S^1 \times S^2 \to M$. Does it follow that $M \cong S^1 \times S^2$? I know that $\pi_1 M$ is cyclic since $f$ must be surjective on $\pi_1$.
Not necessarily. There is a degree one map $S^1 \times S^2 \to S^3$ given by collapsing the $2$-skeleton to a point. Or alternatively, you can construct a degree one map to $S^3$ in this way.
An addendum to Goa'uld's answer: If $M$ is a closed oriented connected 3-manifold which admits a degree 1 map $S^2\times S^1\to M$, then either $M$ is $S^3$ or $M=S^2\times S^1$. See this paper where it is proven that nontrivial lens spaces cannot be targets of degree 1 maps from $S^2\times S^1$.