You cannot deduct mileage for any trip from or to your home. Those trips are commuting, which is not deductible. You can deduct, as a business expense, mileage for trips between work locations - from your W-2 job to a client, or from one client to another.
What you have to do is keep a detailed contemporaneous log of every trip, showing the date, starting point, destination, and mileage. When you prepare your tax return at the end of the year you will have to enter your total mileage for business (the deductible miles), commuting, and personal use. Those three should add up to the total mileage on the car for the year. You do not have to have a home office for this.
You might be able to deduct the mileage for trips between home and a client (but not your W-2 job) if your home qualified as your "principal place of business" for the private tutoring work. But you probably don't meet the requirements for your home to be your principal place of business. First of all, you would have to have a space in your home that is used exclusively for the business. There are exceptions to the exclusive use rule for storage of inventory and running a daycare center, but those don't apply to you.
