This page lists some examples of Adem relations, sorted by degree. The general formula is that when $i \lt 2j$, $$\mathrm{Sq}^i \mathrm{Sq}^j = \sum_{k \in \mathbb{Z}} \binom{j - k - 1}{i - 2k} \mathrm{Sq}^{i + j - k} \mathrm{Sq}^k,$$ with the understanding that $\binom{n}{m} = 0$ if $m \gt n$ or $m \lt 0$, and $\mathrm{Sq}^k = 0$ if $k \lt 0$ (so in practice $k$ ranges from $\max(0, i - j + 1)$ to $\lfloor \frac{i}{2}\rfloor$).

I learnt the following "derivation rule" from Eric Peterson: all the Adem relations follow from $\mathrm{Sq}^{2n - 1} \mathrm{Sq}^n = 0$ and asserting that the map $\mathrm{Sq}^{n} \mapsto \mathrm{Sq}^{n - 1}$ is a derivation.

"i don't know who this version is due to. i learned it from sean tilson who learned it from bob bruner to whom many mysterious things seem obvious" — Eric Peterson

Edited to add: Bob Bruner in turn learnt it from Marcel Bokstedt who attributes it to Kristensen's On Secondary Cohomology Operations

A useful interactive calculator can be found at Alex Kruckman's webpage.