New here and also very new to SharePoint.
I am looking for the best way to manage user and site alerts without deleting them. i.e making changes to the current alerts.
No, another user (or administrator) can't edit another users alerts:
Ref: Alerts Overview
What you can do is this
Start by going to your list where you have the alert and chose the List tab. In the Share & Track section you find the Alert Me icon and text. Select the arrow to bring up the menu and chose "Manage My Alerts"
Select the alert you want to manage, in my screenshot it's called "My Alert".
Then you get the whole list of things you can do to manage your alert. Don't forget to save when your're finished.
To view or delete other users alerts, you need to go to