I would like to know if the following is known or where I can find some information to ask the following question.
Let $f\colon M\longrightarrow S^2$ be a non-constant smooth map between smooth manifolds.
I'm wondering how the image of this map can be and if assuming that the dimension of $M$ is large enough I can guarantee that the image of it contains a circle.
Its image need not contain a circle. Here is a cheap construction: First map $M$ surjectively to $[0,1]$ and then compose it with a map $[0,1]\rightarrow S^2$ which does not contain a circle.
BTW: A good question to ask is how many homotopy classes of maps $M\rightarrow S^2$ there are. This has everything to do with the Pontryagin-Thom construction. A good reference is the wonderful book of Milnor: "Topology from the differentiable viewpoint".