Here are a couple of TurboTax FAQs that will help you report self-employed income without a 1099-MISC and types of self-employment expenses that are deductible.
As for your mileage question, the mileage from your regular job (W-2) to your mystery shop is deductible. Same for mileage from mystery shop to mystery shop. Mileage from a mystery shop to your home or from your home to regular job (W-2) is commuting, so it is not deductible. I am also attaching a screenshot from IRS Publication 463 that shows you what mileage is deductible as well as a link to Publication 463.
How do I report self-employment income from cash, personal checks, or credit card?
https://ttlc.intuit.com/replies/4801198
What self-employment expenses can I deduct?
https://ttlc.intuit.com/replies/5559255
IRS Publication 463:
Travel,
Entertainment, Gift, and Car Expenses IRS Publication 463
