This is a quick rundown on how to get started (or improve) your live streaming at various price points. I am sticking close to products I have actually used and/or have experience with. There are no doubts other options out there for each.
The basics to keep in mind is Audio > Video. Invest in audio first and get that right. It is also true that audio equipment is the most portable between the setups and doesn't tend to go out of date as quickly. Second, keep an eye on where you want to go. You may be happy with a single camera angle now but where do you want to be in the future?
Assuming you have a Smartphone with a reasonable camera, start with that. I also assume you won't be using your normal sound system and will be using a microphone. Basics you will need are:
A tripod adapter for your phone.
Portable battery (you don't want to run out of charge during your stream).
First decide if you will be mic'ing just one person at a time or a group. If it's just one person a lapel mic will work best. Otherwise you want a directional mic that listens on a certain direction.
For lapel the choice is really wired where you are physically connected to your camera or wireless. Do NOT buy a cheap $20 mic, you won't be happy with it. For wired get a Rode SmartLav+ ($100), for wireless get Rode Wireless Go ($200).
For directional mics the choice is mics that attach to your phone or ones that are seperate. Ones that attach are simple and the best place to start. If you plan develop your system more consider a detached mic you can move to other equipment down the line. for iPhone users you want a Shure MV88. For phones with a headphone jack try a Rode VideoMic Me.
For a detached directional mic look at the Rode NTG2 (or NTG4, or really anything in the series). These mics need power to work. The NTG2 can use either a AA battery or phantom power. With your phone you'll use the battery. You will also need a cable to plug it in. The mic uses the XLR standard while nearly all smart phones use the 3.5mm TRRS standard (Note, TRRS isn't the same as TRS, the difference matters!)
$$ (500 - 1500)
The shift here is you are moving to using a computer or something else to stream your content. So you'll need a camera and something to convert the video output into a stream. Nearly every camera will output over HDMI, so that's what we'll assume here. Again, think not only about now but the future. It might be worth investing more in something you can grow with it.
First you'll need a camera. Either get something that will work and is inexpensive or invest in something more professional. For cheap but effective I like the Canon Vixia R800. If you want something to anchor a future multi-camera system go with the Canon XC10 (or XC15).
Second you'll need a streaming device. If you are using a PC you'll need a capture box to get the HDMI signal into your computer. You will find many of these labeled as "Game Capture" devices because that's there dominate use. I have not used any of these products but pretty much anything you find on Amazon should work.
The other option is something that will take your HDMI signal and do the streaming for you, no computer needed. Basic but affective option is the Teradek Video Pro or Mini. If multicam is in your future than you want a SlingStudio HUB. The Teradek is fully self contained while you'll need an iPad or Mac Computer to drive the SlingStudio.
Do NOT rely on the on camera mics. You will need either a detached mic like the Rode NTG2 listed above or a feed from your soundboard. The question is when will your sound join with your video. Simplest option is to use the microphone in on your camera. Here you will need an XLR to 3.5mm TRS adapter. If you use a Teradek or SlingStudio you can also use there audio inputs to mix the video and audio together. Which ever is easier.
Here is where you move to a multi-camera setup built around the SlingStudio HUB. The hub comes with a single HDMI input that you can use for one camera. If you have a computer running slides you want to capture that signal too so you can do fun Picture in Picture things! You may also want a seperate camera capturing a second angle. To get those extra inputs (up to four) you'll need a SlingStudio CameraLink for each (don't let the name fool you, it can take any HDMI source including your computer, dvd player, etc). You can switch between the sources with the SlingStudio iPad app. The Hub will create its own WiFi network that the iPad and Camera links attach too. So you do not need a seperate WiFi network, though you will need an internet connection for the Hub to attach to (wired or WiFi). You can also connect an iPhone, iPod, or iPad as a video source to the Hub :)
What We Do
At our church we have a SlingStudio Hub with two CameraLinks. We have a Canon XC10 as our primary camera and a Canon R800 as a secondary wide shot. We have our presentation computer connected as well as an iPod with an attached Rode VideoMic Me as our "crowd shot" camera that faces the congregation. Normally we take a dedicated AUX feed from our Soundboard for audio into the Hub's audio in.