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 $\mathbb{R}$ $<$ $\mathbb{C}$.

No, it is not correct. Both the real numbers $\mathbb{R}$ and the complex numbers $\mathbb{C}$ have the same cardinality, $\mathfrak{c}$, which is the cardinality of the continuum. See this wikipedia page.
As a general rule, if $A\subseteq B$ then you can say that $|A|\leq|B|$, but if $A\subset B$ (that's a strict subset relation) you cannot conclude that $|A| < |B|$.
