Is it correct to say that $\mathbb{R}$ has fewer elements than $\mathbb{C}$ if both are infinite? My math teacher said that. I disagreed, but he said that I was wrong. But I'm not convinced - is it really right? Please notice that I'm not talking about $\mathbb{R}$ $⊂$ $\mathbb{C}$,  but $\mathbb{R}$ $<$ $\mathbb{C}$.

$\mathbb R$ is smaller than $\mathbb C$ ... ??? In the sense of cardinality, NO.  (As explained in other answers.)  In the sense of Hausdorff dimension, YES.  What sense did your teacher mean?   
I'm sorry, but we often find when a student reports what a teacher says, some details may get distorted or even omitted.  Unless the teacher comes here and makes a defense, we cannot know what is going on.  Or, even better, if the student talks to the teacher (what an idea!) and finds out what was intended.
