A couple mnemonics I find helpful: When we write \$$f\dashv g\$$, the left adjoint \$$f\$$ is on the left, and the right adjoint \$$g\$$ is on the right.

Also, in the important relationships defining adjoints, the left adjoint appears on the left side of \$$\leq\$$, and the right adjoint appears on the right side. For example:
$f(a)\leq b \iff a \leq g(b).$
The \$$f\$$ appears on the left-hand side of the first inequality, and the \$$g\$$ appears on the right side of the second. And they're still on their correct sides if we write the two inequalities in the other order, as in
$a \leq g(b) \iff f(a)\leq b.$

The rule of thumb for other important inequalities, like \$$a \leq g(f(a))\$$ and \$$f(g(b)) \leq b\$$, if you look at which function is on the outside: the right adjoint \$$g\$$ is on the right side of \$$\leq\$$ when it's on the outside of the composite, and when the left adjoint is on the outside it's on the left side. They need to be there in order for the defining relationship to translate these two inequalities into the always true statements \$$f(a)\leq f(a)\$$ and \$$g(b)\leq g(b)\$$.