I tried to solve this problem. I think I solved it but I'm not really whether this is a correct proof.
Problem: All matrices $A\in \mathcal{M}_{4}(\mathbb{R})$ with eigenvalues of $0,1,2,3$ are similar? Yes/No?
My solution: We have four eigenvalues, so we have four linearly independent eigenvectors. These four distinct eigenvalues correspond to four distinct eigenvectors. So, we conclude that we have a basis for $\mathbb{R}^4$ consisting of eigenvectors of $A$. So all matrices $A$ are similar to diagonal matrix $\operatorname{Diag}(0,1,2,3)$. So they are all similar. I think my proof is correct but I'm not sure. Hopefully someone can verify this. Thank you!
I think your argument is fine. You might phrase it more clearly in the following manner: all $A\in \mathcal{M}_4(\mathbb{R})$ with (distinct) eigenvalues $0,1,2,3$ are similar to the diagonal matrix $$ D= \begin{bmatrix} 0&&&\\ &1&&\\ &&2&\\ &&&3 \end{bmatrix}.$$ Similarity is an equivalence relation on $\mathcal{M}_4(\mathbb{R})$, so by transitivity, we get that any two matrices $A,B\in \mathcal{M}_4(\mathbb{R})$ with distinct eigenvalues $0,1,2,3$ are similar.