There is often a LOT of confusion around copyright and licensing, particularly when it comes to the world of broadcast. I'll preface all of this by saying, I'm not a lawyer, and I don't play one on TV... but here is my understanding of your situation as a church that wants to play on TV.
If you are synchronizing video with audio, in other words, using copyrighted music as a music bed for video, you need a synch license. This includes your show open, any music videos you do, or even a mission team report that has a music track underneath.
Synch licenses are, frankly, a pain in the neck. They used to be easy, because the Harry Fox agency had an online tool that let you get them quickly. They are out of that business now. The still do mechanicals and other license types.
Another online service, MusicServices.org offers the same service, and they seem to have a pretty solid library of contemporary Christian tunes. That's a huge help. They also do mechanicals and other types of licenses online.
If the song you want isn't in their library, you have two options. First, you can contact the individual rights holder to negotiate them. This takes time. Lots of time.
You can also hire Church Copyright Administrators to do this for you. It's easy, but because of their minimum project fees, it can be rather pricey, if you're just trying to get a quick video done for the youth.
An easier path is to purchase royalty free tracks for these uses. There are dozens of options, including Killer Tracks, Music Bakery and many others. For a single fee, you get use of the track for whatever you want, forever.
Now... none of this applies to web streaming. To be legal for web streaming, you need the appropriate license from ASCAP, SESAC and BMI. The easy way is to purchase a blanket WorshipCast license from Church Copyright Administrators. You have to report your music use on a quarterly basis, but it's easy to do online.
But wait, there's more. If you're offering the service for DOWNLOAD... the ChurchCA blanket doesn't cover you. You're back to dealing with ASCAP, BMI and SESAC directly.