I'm introducing myself to Complex analysis and Möbius transformations and I read that Möbius transformations map circles and lines to circles and lines.
Are there any other functions that are not Möbius transformations but they can map circles to circles?
If I know that $f(z)$ maps a circle to another circle, can I assume that $f(z)$ is a Möbius transformation?
On
I presume you talk about analytic maps. But even then you may take products of Möbius transformations which also maps $S^1=\{|z|=1\}$ to itself (1-1). Such transformations are called: Blaschke products
If you do not require 1-1 then you also have maps like $z\mapsto z^p$ and if you require analyticity only in a neighborhood of $S^1$ there are many more.
On the other hand a map that always maps any circle or line to a circle or a line is either a Möbius transformation (whence meromorphic) or a Möbius transformation composed with complex conjugation. Perhaps this is more what you are after... (and a proof is not that difficult)
To elaborate on sometempname's comment: if $f(z)=\overline{z}$, then for the circle $|z-a|=r$, we have $$|f(z)-\overline{a}|=r,$$ so the image of a circle is a circle. Similarly, the image of a line is a line, so this will have the desired property but not be a Mobius transform.