I am independent contractor delivery driver. How do I keep track of my gas expenses and mileage? What do I need at the end of the year to prove my expenses?

Simple. You write it down. Just keep a log book for actual valid miles driven and separate personal use miles from business miles. Keep all receipts too.
If the vehicle is used for less than 100% business use, it can be rather difficult (if not impossible) to prove actual expenses for business miles driven. That's why it's generally better to take the per-mile deduction. You actually get more of a deduction that way in many situations.

Thanks for your response. What proof do I provide for the mileage I'm reporting?  Also, will I not report gas expenses then?

Your proof is your logbook that you keep as you go.

There are two ways you can calculate the deduction for using your car for business. You can (1) keep track of each work-related mile you drive and multiply the total by the IRS standard mileage rate, OR (2)  you can save the receipts for the actual cost of gasoline and oil.

Thank you

I just started delivery driving last year. I also ended up getting quickbooks. You can log all your mileage with quickbooks, and choose which is personal and which is for business. Super easy. No need to keep a big book either. You can also attach your receipts to your bank transactions, this will log gas.