how to calculate $q$-expansion coefficients of a modular form? Can you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused.

Finding Fourier coefficients can be also be achieved by the method of Rademacher sums and series, as outlined in this paper by M. Cheng. 
The general idea is we can construct a symmetric function from a non-symmetric function $f$ by summing over the images under the desired group. Poincaré considered instead summing over a function $f$ already invariant under a subgroup and sum over representatives of cosets of the subgroup fixing $f$.
We could take as an example, $f(\tau) = e^{2\pi im\tau}$ for $m \in \mathbb Z$ and $\tau \in \mathbb H$, which is invariant under the subgroup $\Gamma_\infty \in SL_2(\mathbb Z)$ of upper-triangular matrices, leading us to consider the sum,
$$ \sum_{((a,b),(c,d)) \in \Gamma_\infty \setminus SL_2(\mathbb Z)} f\left( \frac{a\tau +b}{c\tau + d}\right) \frac{1}{(c\tau +d)^w}$$
for even $w$. The study of these sums of this type for $w = 0$ is first attributed to Rademacher. The details beyond this point are quite elaborate, but we are able to obtain expressions for the $n$th Fourier coefficient in many cases.
As an example, for the $j(\tau)$ function, the Rademacher coefficients correspond to Fourier coefficients (modulo some massaging), which are given by,
$$c_{\Gamma, 1 ,0}(-1, n) = \sum_{c,d } e \left(\frac{a+nd}{c} \right)n^{-1/2} \frac{2\pi}{c} I_1 \left( \frac{4\pi}{c}n^{1/2}\right)$$
for $c,d$ coprime, $0 \leq d < c$ and $c>0$, which recovers the $n$th Fourier coefficient, which is related to a Rademacher sum, due to Petersson, 
$$P^{[\mu]}_{\Gamma, \psi, w} = q^\mu + \sum_{h\nu +\alpha \in \mathbb Z, \nu \geq 0} c_{\Gamma, \psi, w} (\mu, \nu) q^\nu$$
where $\Gamma(N)$ is the principal congruence group, $\psi$ is the multiplier, $w, \mu \in \mathbb Z$ with $w > 2$ and even. The $j(\tau)$
function coefficients are extracted for the case $\Gamma = SL_2(\mathbb Z)$, $\psi = 1$ and $w = 0$, with $\alpha = 0$ and $(\mu,\nu) \in \mathbb Z \times \mathbb Z$. 
