How do I know when they've received my taxes by mail?

When you mail your return to the IRS allow about 4 weeks then you can start checking the status on the IRS site:

To track your Federal refund:   

https://www.irs.gov/refunds

To track your state refund:    

https://ttlc.intuit.com/questions/1899433

On the IRS site you are also offered the use of an app called IRS2Go