Apply for SSN within 1 week of US entry on K1 visa. Deadline is until day 76 after US entry, but you would want to apply for SSN as soon as possible, in case there are delays with SSA verifying the applicant's immigration status.
Note that even if the K1 visa holder gets SSN, they are not legally authorized to work until they get their EAD or GC.
Check the thread below for more suggestions on the order of tasks after arriving on K1 --