Write down a sentence $\phi$ that for any domain $M$, if $M\models \phi$ then $\vert M\vert <4$
So I want a sentence that says there are less then $4$ elements.
I dont think this sentence works. $\exists y\exists r\exists z\forall x (x=y \vee x=r\vee x=z)$
When I use quantifiers like this $\exists y\exists r\exists z$ does it imply that $r,y,z$ are distinct? Because if it does then I believe my sentence says there are exactly $3$ elements which isn't what I want.
The other sentence I thought of would be $\neg \exists y\exists r\exists z\exists x(x\neq y\wedge x\neq z\wedge x\neq r\wedge y\neq z\wedge y\neq r\wedge z\neq r)$
Which I believe says there aren't 4 distinct elements which should imply there are less then $4$.
Your sentence is almost correct—it is not implied that variables with different names are distinct.
The reason it is only 'almost' correct is because $M$ might be empty, in which case any sentence of the form $\exists y \cdots$ is false. You need to account for that case as well.