The following exercises are relevant to your question (all matrices are assumed to be $n\times n$ square matrices and $V$ is a vector space of dimension $n$):
Exercise 1: Let $A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ and let $B=\begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}$. Prove that:
(a) $AB\neq BA$ and
(b) $A^{-1}BA\neq B$.
Exercise 2: Prove that if $A$ and $B$ are arbitrary matrices and $B$ is invertible, then $AB=BA$ if and only if $B^{-1}AB=A$.
Exercise 3: Let $I$ denote the identity matrix. If $A$ is similar to $I$, then prove that $A=I$.
Exercise 4: Prove that the relation $\equiv$ on the set of all matrices defined by the rule $A\equiv B$ if and only if $A$ is similar to $B$ is an equivalence relation.
Exercise 5: Describe the matrices that constitute an equivalence class consisting of exactly one element. Does there exist an equivalence class consisting of exactly two elements? Prove or give a counterexample. More generally, if $n$ is a positive integer, does there exist an equivalence class consisting of exactly $n$ elements? Prove or give a counterexample. Finally, does there exist an equivalence class consisting of a countably infinite number of elements? Prove or give a counterexample.
Exercise 6: Prove that if $A$ is a matrix such that $AB=BA$ for all invertible matrices $B$, then $A=cI$ for some scalar $c$ where $I$ is the identity matrix. Deduce that if $A\neq cI$ for any scalar $c$, then there exists a matrix $B\neq A$ such that $B$ is similar to $A$. Hence similar matrices that are not equal exist in abundance.
Exercise 7: Let $A$ be a matrix and suppose that $A$ is similar to a diagonal matrix $B$ where all the diagonal entries of $B$ are equal. What can you deduce about $A$?
Exercise 8: If $A$ and $B$ are diagonal matrices, then prove that $AB=BA$. If $B$ is invertible, deduce that $B^{-1}AB=A$.
Exercise 9: If $B=C^{-1}AC$ for some invertible matrix $C$, then prove that $p(B)=C^{-1}p(A)C$.
Exercise 10: Let $A$ be a matrix and suppose that $A$ is similar to a diagonal matrix. Let $\lambda_1,\dots,\lambda_n$ be the diagonal entries of this diagonal matrix (repeated according to multiplicity). If $p$ is the polynomial defined by the rule $p(x)=(x-\lambda_1)\cdots (x-\lambda_n)$, then prove that $p(A)=0$.
Challenging Exercises:
Exercise A: Let $A$ be a matrix with complex entries. Prove that there is an invertible matrix $C$ with complex entries such that $C^{-1}AC$ is an upper triangular matrix.
Exercise B: A matrix $A$ with real or complex entries is said to be self-adjoint if $A$ is equal to its conjugate transpose. (The conjugate transpose of $A$ is the matrix obtained by taking the transpose of $A$ and then taking the complex conjugate of each of the entries of the transpose of $A$.) Prove the spectral theorem; that is, prove that there is a unitary matrix $U$ such that $U^{-1}AU$ is a diagonal matrix. (Let us recall that a matrix is unitary if its columns form an orthonormal tuple in $V$.)
Exercise C: A matrix $A$ is said to be normal if $AA^{*}=A^{*}A$ where $A^{*}$ denotes the conjugate transpose of $A$. Prove the complex spectral theorem; that is, prove that if $A$ is a normal matrix with complex entries, then there exists a unitary matrix $U$ such that $U^{-1}AU$ is a diagonal matrix.