Cartesian product of $S^1$ and $S^2$ Can anyone help me imagine $S^1 \times S^2$? I understand that $T^2=S^1 \times S^1$, but I don't know what to do with the spheres. I am not even sure if it's in 3D. Thank you!

This is a good example of a 3-dimensional manifold that cannot be embedded into the Euclidean space $\mathbb R^3$.
The best way you can think of it is as a circle such that it has a sphere associated to each and every point of it.
I bet you already know how to parameterise both the circle $S^1$ (call the parameter $\rho$, for instance) and the sphere $S^2$ (say the parameters are $\varphi$ and $\theta$). Then, you can just put your parameterisations together and get 
$$\Phi:(\rho, \varphi,\theta)\mapsto \Phi(\rho, \varphi,\theta)\in S^1 \times S^2.
$$
If you are concerned with regards to its embedding into an Euclidean space $\mathbb R^n$, the best I can offer you is $n=5$. That would come with the $\Phi$ from above, which embeds $S^1$ into $\mathbb R^2$ and $S^2$ into $\mathbb R^3$.
