Can I set up different permissions for each employee?

Yes.  In the Employees module, you can modify employees permissions by going to the edit screen.  On the Permissions tab there are two check boxes.  By checking the Manager permission, the employee will have access to all of the modules in Fishbowl Time & Labor.  By checking the Mobile App permission, the employee will have access to clock in/out on the mobile app.  To see more about setting up an employee go here.

permissions

Employees may also be allowed to edit their time cards. Under the Settings menu, click on Time Clock and check the appropriate boxes for allowing employees to create/edit time.  Having this checked will allow any employee with a login to edit their own time cards.

Time Clock settings