I am having a hard time understandig why one could write (for smooth functions with compact support):
$$\left(\int |f|^p\right)^{1/p}+\left(\int |g|^p\right)^{1/p}\leq A \left(\int |f|^p+\int |g|^p\right)^{1/p} $$
Evan's passage I am referring to is this one:
Short answer: by equivalence of the norms in dimension $2$.
Details. Let $a = \|f\|_{L^p}$ and $b= \|g\|_{L^p}$. Then by equivalence of the norms in finite dimension, there exists $A>0$ such that $$ a+b = \|(a,b)\|_1 \leq A\,\|(a,b)\|_p = A \,(a^p+b^p)^{1/p} $$ and this is exactly your inequality.
Quantitative version. One can be more precise about the constant. Since $p\geq 1$, the function $x\mapsto x^p$ is convex and so $(\frac{a+b}{2})^p \leq \frac{a^p+b^p}{2}$, that is $$ a+b \leq 2^{1-\frac{1}{p}}\, (a^p+b^p)^\frac{1}{p} $$ so the constant is $A = 2^{1-\frac{1}{p}}$.