Hello @nicol3
Pages 16-17 of the manual explains how the hard drive would be mounted.
http://h10032.www1.hp.com/ctg/Manual/c06140594
I dont know if the mounting screws are included with the Obelisk, but the tray/caddy should be there according to the images in the manual.
If the screws are not stashed in a package when you purchased the pc, you will want to buy some. Although double sided tape would work also.
Then you would connect a sata cable that you provide, as you pointed out. There will be a sata power cable from the PSU that also needs to be connected.
Once done, boot up the PC. If you only want to use the drive as data storage, there is nothing special that needs to be done.
It may or may not be usable yet. You can go to Disk Management and intialize and format the drive if needed.
https://www.easeus.com/partition-manager-software/initialize-a-hard-drive-ssd.html
Is this what you needed to know?