@Lelith Walker and @John Morgan phone apps are a great resource. check out "invest home pro", "Joist", "home inspection checklist", they are pretty cool. 

I did some things right with that first rental (scheduling an open house instead of individual appointments, for example, and screening applicants) but there's definitely room for improvement.

I grabbed a couple addresses off Realtor.com and used it to find the BBL, but from my phone it seemed like a pretty pointless exercise. 

After a few phone calls title insurance paid us back the $13,000.- we paid for the house. 

When it comes to screening prospective tenants for income, do folks rely on a service for this or by calling on the phone?

I'm confident some of them would not object to a free phone conversation with you, especially if there is a strong likelihood that you'll ask them to do some legal work.

The first couple of things you have to do is to set up your company as a corporation of some type to protect you personally and find/get a local Team put together that will return your phone calls.  

What's the point of screening and vetting tenants if they can just slip in once their family qualifies?

Honeywell has a nice thermostat at the big box stores around $225 that is wifi and has a phone app that is really nice. 

Does not answer through phone, and sometimes takes over 3 days to respond through email.