Is a modular form $f$ of weight $k$ with respect to $\text{SL}_2(\mathbb{Z})$ always a modular form to a congruence subgroup $\Gamma$ (for example $\Gamma_1(N)$)?
If the transformation law $f|_k\gamma=f$ holds for any $\gamma\in \text{SL}_2(\mathbb{Z})$ then it holds also for any $\gamma\in\Gamma\subset \text{SL}_2(\mathbb{Z}).$ But is $f$ holomorphic at all cusps of $\Gamma$?
Thanks in advance.
Yes, a modular form on $\text{SL}_2(\mathbb{Z})$ is a modular form on any congruence subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$. It's quite clear that the transformation law holds, as you've mentioned.
Suppose that $\alpha \neq \infty$ is a cusp of $\Gamma$. Then since $\alpha$ is equivalent to $\infty$ in $\text{SL}_2(\mathbb{Z})$, i.e. there is some $\gamma \in \text{SL}_2(\mathbb{Z})$ such that $\gamma^{-1} \alpha = \infty$, the behaviour of $f$ at $\alpha$ is the same as the behavior of $f(\gamma z)$ at $\infty$. Since $f(\gamma z)$ is well-behaved at $\infty$, you get that $f(z)$ is well-behaved at $\alpha$, and so $f$ is holomorphic at $\alpha$.
Generally, each modular form on a congruence subgroup $\Gamma' \supset \Gamma$ is also a modular form on $\Gamma$. The forms coming from these larger congruence subgroups are often called "oldforms", while those forms first appearing on $\Gamma$ itself are called "newforms." When studying modular forms of level $N$, it is usually good to look at the newforms of level $N$ as the oldforms have slightly different (and often slightly better) behaviours.