I just purchased a 1988 Hi-Power Standard with adjustable sights. Have not fired it yet. At what distance should I zero the sights? Thanks!

If height over bore is at .75" sight in for 10 yards then check it to see it is hitting on at 50 yards.  Will give you point of impact within an 1inch of line of sight out to 60 yards if you could actually shoot Point of Aim= Point of impact to same exact point out to 60 yards or in other words around .001 inch group.

If this is your first handgun sight it in using padded rest for your arms.
Shoot 50 rounds a day for 10 days in a row.  Start at 5 yards on K5 type target and move back to 10, 15 and 25 yards as groups get smaller.
Dry fire at night in a room later where there is absolutely no ammo in room.  
Many "gun cleaning" accidents are actually someone who decides to dry fire just one more time before exiting the room because they had ammo in the room and reloaded the gun already...........

I've yet to have an accidental discharge of any kind.

I always tell people to clear weapon by dropping the magazine and pulling the slide to the rear cycling it 3 times and then look in the chamber before releasing the slide.  They will see the second round come out of their ejection port and figure out they forgot to eject the magazine sooner or later.

At whatever distance you normally shoot at.