It's worth distinguishing between the prime coefficients $a_p$, and the coefficients $a_n$ for general $n$. Let's look at $a_p$ first.
Firstly: for elliptic curves, it is fairly easy and elementary to prove that $|a_p| < 2\sqrt{p}$ (Hasse's inequality). This should be in any decent textbook e.g. Silverman. And this is the best bound you're going to get, because for any fixed elliptic curve the set $\{ a_p / (2\sqrt{p}) : p\ prime\}$ is dense in $(-1, 1)$, and in fact obeys a very specific distribution (the Sato-Tate distribution).
For general modular forms of weight $k$, the "right" bound is $|a_p| < 2p^{(k-1)/2}$, but this is a very deep theorem (it follows from Deligne's work on the Weil conjectures). There is an easy elementary argument that gives $a_p = O(p^{k/2})$: this is in Miyake's book (corollary 2.1.6 if I recall correctly). By purely analytic methods you can push this a bit further, e.g. Rankin proved $a_p = O(p^{k/2 - 1/5})$ if I remember correctly, but you can't get the "right" bound this way.
For general $n$, the relations giving the $a_n$ in terms of the $a_p$ mean that the above bound $a_p = O(p^{(k-1)/2})$ turns into something like $a_n = O(n^{(k-1)/2})d(n)$, where $d(n)$ is the number of divisors of $n$; this is $O(n^{(k-1)/2 + \epsilon})$ for any $\epsilon > 0$ (but it is not $O(n^{(k-1)/2})$).