I just received payment on my first invoice. Does it transfer automatically into my connected bank account, or is there a step I need to take to make that happen?

Hello,

The actual payment will transfer automatically into your bank account that you have setup to receive payments. (Double check your merchant account settings to make sure its the correct bank account)

Then when the payment posts in your bank account, it will download the transaction in your bank feed in QB Self-Employed.  Mark it as business income and that will make it count as income that reflects on your profit and loss report and estimated self-employed tax.