How will my earnings from Rover need to be reported in my taxes? Will I get a 1099?

No, you will not get a 1099 but the earnings are reportable anyway. You are an independent contractor and would file a Schedule C.

A good rule of thumb is to set aside 15-30% of your earnings to cover self-employment taxes that you will owe when you file your income taxes. You can either put it in a savings account or use EFTPS.gov to actually pay in the money (considered estimated tax payment).