I'm going to assume your measure space is $\mathbb{R}$ with Lebesgue measure.
Consider $f_n=n1_{[0,\frac{1}{n}]}$ where $1_A$ is the characteristic function of the set $A\subset \mathbb{R}$. Then $f_n \to f=0$ pointwise but
$ \liminf\int_{\mathbb{R}} f_n(x)dx=1 > 0= \int_{\mathbb{R}} f(x)dx $
(You can even take $f_n, f\in C^{\infty}(\mathbb{R})$ so it's not a regularity issue).
In other measure spaces it might still be false: For example in $\mathbb{N}$ with counting measure take $f_n(m)= 1$ if $m=1,n$ and zero otherwise then $f_n \to f$ pointwise, where $f(1)=1$ and is zero otherwise, and
$ \liminf \int_{\mathbb{N}} f_n(m)dm = 2 > 1 = \int_{\mathbb{N}} f(m)dm $
So no, in general only one inequality is true in Fatou's lemma.
With the edit it's still not true: Take $g_n=-f_n$ as above. You could put $|f_n|\leq f$ but then this is just the dominated convergence theorem.