In principle, as you point out, showing that a number $r$ is rational is easy. All we need to do is to exhibit integers $a$ and $b$, with $b\ne 0$, such that $a=rb$.
Proving that a number $x$ is irrational is in principle, and often in practice, much harder. We have to show that there do not exist integers $a$ and $b$, with $b\ne 0$, such that $a=xb$. So in principle we have to examine all ordered pairs $(a,b)$ of integers, with $b\ne 0$, and show that none of them can possibly "work."
This in principle involves examining an infinite set. That cannot be done by simply exhibiting a pair of integers, like in a proof of rationality. Sometimes, as in the case $x=\sqrt{2}$, there is a relatively simple proof of irrationality. But all too often, like in the case of the Euler-Mascheroni constant $\gamma$, no proof of irrationality is known, despite the fact that considerable effort has been expended trying to prove that $\gamma$ is irrational.