Hi LaManglana!

You’ve outlined a few different questions here but I’ll break them down into two answers that should address everything. And yes, you are on the right path by using a Teams plan. 

• “I need a main calendar… and some ‘personal’ calendars for each user.” and “if someone books an appointment… the date is no longer available”
  - On a Teams Plan you can have as many users as you want. Each can have up to six calendars connected per user. The way Calendly works is that it sits on top of your existing digital calendar(s) and schedules appointments when there is availability on one of those calendars. It can only add and subtract the event types that you set up. Once it has added one of those event types to the synced calendar, that time slot will no longer be available for Calendly to book. Coordinating multiple calendars for one user should be done via linking them by whatever calendar client you use, whether that be Google, Microsoft, etc. Most calendar clients offer the ability to automatically add events from one calendar to another based on automation rules you set up. Again, Calendly won’t have any power to control those rules or settings. It simply sits on top of existing calendars.
• “I need each user to be able to manage their appointments without being able to see the appointments that of] user”
  - User shouldn’t be able to see each other’s appointments if they have their visibility settings setup to ensure privacy on each of their individual calendars. All that Calendly will see, and by extension an invitee using a booking page, is that there either is or isn’t time available. No details of prior meetings will be shared with invitees on booking pages. Additionally, on a Teams Plan, an administrator has extra power to create event types for the team and can control a myriad of settings. I’ll leave more details about Team Plans below. 

More resources:
How Calendly Works

New User Guide

Working with multiple email accounts and calendars

Teams Plan

Setting Up A Teams Page

Using Calendly As A Team