What doesn't constitute as a company costs when driving?

Use company card for any mileage that you are driving for business purposes.

It'll make tracking and end-of-year reporting much simpler.

You may not, however, use the mileage you drive to and from your office (the IRS labels that as commute mileage) as a tax-deductible business expense.

If you're looking for deductions for your business and using a personal vehicle, keep track of mileage. Get a mileage app, Mile IQ comes to mind. (Don't ask me if it's any good, find one you like) Track personal miles and business miles, it comes in handy for the tax preparer. The 2018 rate is 54.5 cents a mile and it adds up quick. There are a few places where keeping all your receipts is advantageous when it comes to mileage, but that's unlikely. The 54.5 cents includes Fuel, Maintenance, and Depreciation. So any expenses associated with those three categories cannot be deducted when taking the standard mileage rate on the Sch C, F, or E. The standard mileage rate is almost always higher.

Do you volunteer? Keep track of those miles. 14 cents a mile.
Going to the Doctor? Keep track of those miles. 18 cents a mile.

Now if you're working a job with a W-2 and your employer is not reimbursing you, you're SOL since 2% misc. deductions are now defunct. That doesn't sound like the case since you have a company card, but it was worth saying.

If you use your personal vehicle for your business, the IRS allows you deduct vehicle related expenses as part of operating your business. You can use one of two methods: the Actual Expenses method, or the Standard Mileage Method. Each method requires different record keeping requirements and can produce varying results. Generally you go with the method that produces the largest tax deduction. The actual expenses method requires you to track and add up all of the money spent on operating your vehicle (gas, insurance, maintenance, etc) and take a percentage of the vehicle's business use. So, if you estimate you use your vehicle 30% of the time for business, you would deduct 30% of those total costs. The Standard Mileage Method requires you to track the total miles driven for business during the year. Multiply the total miles driven by the standard mileage rates published by the IRS for that tax year.