Screening isn't complicated, but it's easy to do inconsistently, which is where landlords get into trouble. Here's the process in order.
Application first, always. Get everyone who'll live there, even minors' guardians, to fill out the same application: income, employer, current address, previous landlord contact, and consent to a credit and background check. Use the same form for every applicant. Inconsistent screening is the single easiest way to end up in a fair-housing complaint, even by accident.
Income verification. Pay stubs or bank statements, not just a stated number. A common rule of thumb is gross monthly income at least three times the rent, but decide your own bar in advance and apply it the same way to everyone.
Credit and background check. There are inexpensive screening services built for individual landlords that pull both at once. Look for payment history, not just a score. A thin credit file isn't the same thing as a bad one.
Call the current landlord, not just the listed reference. A applicant's friend posing as a landlord is more common than people think. Ask specific questions: was rent on time, was notice given properly, would you rent to them again.
Decide against your written criteria, not your gut. Write down your minimum standards before you screen anyone, then apply them the same way every time. That's not just good practice, it's what protects you if a decision is ever questioned.