I drive part-time for Skip the Dishes. I got a warning that I need to pick a province for this. Is this considered to be a perm. business?

Yes. Your Skip-the-Dishes income is considered to be self-employment/business income for tax purposes. You'll need to complete the Business section of your return.

How do we add these income in 2019 Standard edition. I am finding to add these income and cant claim for mileage expense with this edition 

To enter your mileage, using TurboTax Online, click on the “Find” icon in the blue menu and search for “self”. Select “Self-Employment” and then click the “Go” button. The EasyStep interview will eventually bring you to the “Motor Vehicle Expenses Summary” where you can add you mileage and other related information.