@Bryan La
For us, the first step in screening is the initial phone call and in-person interaction when they tour the property. While people will likely be on their best behavior since they're trying to impress you, you'll still be able to get a feeling for how they'll be as a tenant. One thing I like to do at this stage is ask why they're moving, and listen to what they complain about where they currently live. Of course, some complaints will be justified, others not, but if they readily share complaints about where they currently live, they'll likely find similar issues at your property, and might not be worth the headache.
We then have them fill out a standard application. Once we have a completed application, we verify income (by calling the employer and/or with paystubs), verify previous address history (this will show up on the credit check, we ask about any addresses that show up on the credit check but are not listed on their application), and verify identity by requesting a copy of their photo ID to make sure their info matches the application and screening reports.
Next, we run credit, criminal, and eviction reports. For my personal rentals, I use apartments.com's feature for this since it's relatively cheap (~$30, paid by the tenant) and it is pretty comprehensive.
In my opinion, the credit check is extremely extremely important. You want tenants who care about their financial lives so much that they ensure they have a high score because they'll make sure that all of their bills get paid on time and in full, including the rent.
Feel free to reach out anytime if I can elaborate or further clarify anything!