How do I delete an appt from my calendar so I can schedule someone else in its place?

Appointments can be re-scheduled for a different date/time or cancelled, but not deleted. However, you can still book multiple appointments in one time slot.

Try this:

Go to the Calendar tab and change your view (top right) from All Appts to Open Appts.

You can also search for the lead/member you wish to book the appt with and use the Actions button to Schedule Appt.