What type of method would I use if a Uber / Lyft handled actions.. Once they took their fee, payment was made to me once a week. Not sure if this is the Cash Method?

Yes, in most cases you are a cash-basis taxpayer, and you use the same method for your Uber/Lyft business. If you used accrual method, you would know and recognize this.

Report the gross income that was paid to you. The fees that were deducted should be reported as a business expense.