If you mean to move a site from one site collection to another, there is no 100% "safe" way. Easiest: use a 3rd party tool, such as ShareGate. These tools turn this job into a couple of mouse clicks. However, it's still not "safe", as the following can still come up:
- Workflows with hard-coded paths, in email actions for example
- Excel spreadsheets that link to other excel spreadsheets via...hard-coded (not relative) paths.
Without 3rd party tools, you can use the export-spweb powershell cmdlet, and then import-spweb. This can move all content, including versions, although it's trickier to use than the 3rd party tools.
If you're moving a site to a different location in the same site collection, then the "content and structure" page in site settings should work fine.
Admins should be taking backups already, but certainly coordinate with them to be sure. For example, if you run the import operation twice, for whatever reason, you wind up with duplicate copies of all content. So yes, ensure the backup exists, and ensure the restore processes have been tested recently. (but again, that advice applies whether or not you're planning on doing anything to the system)
edit: regarding safety: the 3rd party tools and the export-spweb will both result in copies of the site. So if the copy has issues, just delete it and try again. The risk with these tools is that once you make the copy and send the url out to users, you'll need to either delete the original location or revoke users' permissions so that they don't wind up adding content to the original location.