While the multiplicative order of $10$ modulo $2$ or $5$ does not exist, the multiplicative order of $10$ modulo $p$ for $p\geq 7$ prime is not necessarily $p-1$ ($2$ for $p=11$, $6$ for $p=13$, and not $36$ for $37$).
How do I produce a generalisation to find the multiplicative order of $10$ modulo some arbitrary prime $p$?
There is no formula for calculating the order of an element modulo a prime. The only method is (essentially) to start calculating. (There are shortcuts to speed that up in some cases, but nothing that's really fast or easy.)
Marking this answer community wiki.)