Correct set notation for "all integers which are not multiples of 7"?

I'd generally go with $\{x\in\mathbb{Z}: 7 \nmid x\}$, though this does presuppose familiarity with the $\mid \, \nmid$ notation.

For starters, you should be using curly braces $\{\}$ for sets, and what you have is fine, but generally the clean abbreviation would be: $$ \mathbb{Z}-7\mathbb{Z}$$

This is because $-$ denotes set difference and $7\mathbb{Z}$ denotes the set of integer multiples of $7$.

My preferred notation is the "not divides" symbol $a \nmid b$ what means "a not divides b". But you can achieve the same meaning by different ways, by example:

$$\{n\in\Bbb Z: (\nexists k\in\Bbb Z :n/k=7)\}$$

or

$$\{n\in\Bbb Z:n\not\equiv 0\pmod 7\}$$

But the common notation is

$$\{n\in\Bbb Z:7\nmid n\}$$