Bryan,
This can be done by making a user a Direct Manager, not an Admin. An Admin is provided full access to creating/modifying users and groups as well as Global and System settings (including customizations). I do not believe this is what you want. Instead, make the user a Direct Manager on a Project or simply specify certain users to have Reports To = Bryan (for example). That will allow you to see their information (*Tasks included). Then, you can enable the following system setting...
Time Tracking --> 11.9 Direct managers can approve timesheets. This will allow a non-admin user (i.e. Bryan) to approve/change hours reported by users that report to him.
Sincerely,
Boris