A lot of churches start their live streams with either a smart phone or a single camera connected to computer running encoding software. That's fine to start, but as soon as possible, you should consider making the stream more engaging with multiple cameras and live graphics showing sermon notes, etc.
For inspiration, look not only at bigger churches, but also at other events to watch how they're covered. Sporting events have multiple cameras from multiple angles. So do talk shows. As do live events like large conferences. Even live streams from most corporate events aren't just a single camera, but multiple.
Adding in a switcher
Moving from a single camera to two sources, whether they're both cameras or a camera and a computer, will require additional hardware and/or software. You can't just unplug one source and plug in another. Well, you can, but that's a lot of work for a poor quality result. The right way to do this is with a video switcher.
There are two types of switchers for you to consider. You might opt for software on a computer if you require a more budget-minded approach.
Don't forget to include capture cards and other hardware when you price software switchers. A price of $500 for one of the products might seem like a great deal until you realize that you'll need to spend money on the computer, and other hardware to make it capable of doing what a low-end hardware switcher does out of the box. Additionally, software does crash. A system update could cause your switcher to quit working whether it's on a Mac or Windows box, so consider that, as well.
On the hardware side, there are entry level switchers from Panasonic, Blackmagic Design, Roland, Datavideo and other companies.
If you're just getting into video production, you probably don't need a large two M/E switcher with four auxes, but you might need them in the future. Software switchers and entry-level hardware switchers won't give you a lot of flexibility to send multiple video signals to different places, but the broadcast variety will…for enough money.
There's a third category that's worth noting as well. With tablets and smart phones being so common, it shouldn't surprise you that there are a couple of apps that do the job of a video switcher without capture cards or other traditional hardware. One offering called “Switcher Studio” uses iPhones and iPads as cameras, switching from an iPad. Wowza's Streaming Cloud is similar. Another, called the Mevo, uses a 4k camera to simulate multi-cam shoots by showing only part of the video signal at a time, effectively cropping the video in HD windows from the larger image. Telvue's GoToAir offers similar capabilities. Teredek also makes an app that takes the video over the network from cameras connected to multiple Vidiu encoders.
Each of these approaches has its benefits, but finding a solution that's powerful, inexpensive, and user-friendly is nearly an impossible task. Good switching will pass the signal from the sources without degrading it. Your church's needs will determine which of the rest of the options you need.
Cameras
As for cameras, perhaps a $200 camcorder did the trick by itself, but there are a few features you're going to want to have when you add in a second or third camera to the mix.
Perhaps to start, you'll want a couple of stationary cameras that you can cut between for variety and to show both detail and context. Don't stop there though.
Each camera has the potential of showing a myriad of different shots during a normal church service. A good camera operator will be able to use zoom, focus, framing, and angle to compose a shot that tells the viewer more than a static shot ever could.
No matter how good the camera op, it's hard to create compelling shots with a camera that doesn't have good controls.
Other hardware you'll need
Likewise, good stabilization is a must and a tripod that you can buy at a big box electronics store won't fit the bill. Expect to spend hundreds or thousands on support (depending on your needs).
If space is at a premium PTZs (pan-tilt-zoom remote cameras) are an option to consider. These units are often less capable than a well-trained camera op, but can go places where you just can't put a person.
Another item to add to your list of equipment is an intercom unit (from manufacturers like Telex or Clearcom). There are other solutions, like smartphone apps or DIY projects, but the director needs to communicate with camera ops during your service. Hand signals, text messages, and even two-way radios won't do what you need here. Clear duplex communication is the only thing that works reliably in a live event.
If it's a simple system, getting a couple pro-sumer cameras, tripods, zoom and focus controls, and connecting it to an entry-level switcher could really up your game from a single camera at the back of the room.
If you're doing a more complex system, be sure to get a system integrator involved. You might not know what you need that is missing from this list or that will make it all a lot easier. Multi-viewers, waveform and vectrascope monitors, CCUs (camera control unites), as well as hardware encoders are all items that more complex systems should have that simpler systems don't always require.
Don't forget people
Most importantly, a single-camera live stream might be something that one person can do, but when you add in more cameras, you will be adding more people. For cameras on tripods, this means a camera op for each one. Plan to have someone to run the switcher and perhaps someone to keep an eye on the service as a whole. What was once one-person job for a simple, one-camera stream could easily and quickly become a three or four-person serving opportunity. There are churches that routinely use seven to ten people in various roles, all dedicated live video.
You don't need to worry about getting professional operators in all these seats either, at least not at first. Perhaps an automotive engineer, or a student, or a landscaper would be good at running a camera or calling the shots at your church. As time goes on, perhaps you'll hire freelancers or have a staff, but it's possible to start smaller.
Don't hesitate to find training materials that can help take willing volunteers and transform them into videographers. Investing in people will often pay dividends that are greater than investments in equipment, especially because long-term volunteers will continue to get better and better while equipment ages, breaks, and becomes outdated over time. Don't neglect your people or the best equipment will be useless in their hands.
Moving from a single camera to multiple camera operation adds cost and complexity, but the results are worth it. Instead of having a production that takes away or distracts from the impact of the live event, a multi-camera shoot actually adds to an event, giving a series of perspectives that the live viewer would miss in-person. For people that can't be there, a well-produced stream can make them feel included, elevating the experience from “better than nothing” to truly engaging. That's what makes it worth doing.