You can edit staff contacts via Admin>Personnel>Staff.
From there, browse for the member of staff by clicking on the drop down box.
You can then start typing the staff member's name and select their name as it pops up.
Once their record has opened, click on the "cog" next to their name (above their photo).
This will present you with their "personal details" where you can add/edit details, such as phone number and email address.
If you need to add information for their contacts (emergency contacts, doctor, etc), on their profile front page click on the cog next to Contact Details under the Extra Details header.