Is there any good example about Lie algebra homomorphisms? My textbook gave an example of the trace, but I think to get a better comprehension, more examples are still needed.
Any example will be helpful ~

Let $V_k$ be the space of homogeneous polynomials of degree $k$ on the unknowns $x_1,x_2,\ldots,x_n$. The differential operators $a_{ij}=x_i\dfrac{\partial}{\partial x_j}$ act on the space $V_k$. It is an easy exercise to show that
$$
[a_{ij},a_{kl}]=\delta_{jk}a_{il}-\delta_{il}a_{kj}.
$$
Therefore the mapping $e_{ij}\mapsto a_{ij}$ is a homomorphism of Lie algebras from $gl_n$ to $gl(V_k)$.
Addendum: Here $\delta_{ij}$ is the Kronecker delta, and by $e_{ij}$ I mean the matrix with 1 at the intersection of row $i$ and column $j$ and zeros elsewhere. This is, of course, an example of a simple representation of $gl_n$, but I hope it will be helpful also, when you reach that point in your studies.
