Cameras these days are ridiculously good compared to even 10 years ago, and ridiculously forgiving. I’d recommend you buy a decent webcam (if you have a laptop or other computer handy to run it) or a cheap Android phone (if you don’t.) You don’t have to get service for the phone- the camera and such will all still work, and you can connect it to a computer via USB cable to transfer the video. You can get one of these things for US $50 or so. An advantage of an Android phone is that, if you have service, you can upload to Youtube directly from the phone. You might be surprised at the high quality of today’s cheap cameras, but if it’s not enough for you (almost any phone camera these days can record at a higher quality than YouTube can display) then you can always buy a fancier unit later.
One thing you must have is a tripod. You don’t want to stand around holding a phone to record an entire service, and your viewers don’t want to get seasick from shaky-cam. Image stabilizing programs exist, but using a tripod will make their job that much easier. Tripods are cheap. Get a good one and learn to use it. If you have a good tripod while you’re messing around with recording your church services, you’ll still be able to use it if and when you move up to a more expensive video camera and/or higher-end videography. Yes, that means you should spend more on your tripod than on your first camera.
Lighting is very important, but it’s also somewhat dependent on the space in which the services take place. If your church has a “stage manager” or someone similar who sets up the lighting and sound reinforcement for the services, talk to that person and get their advice. I’ll mention two things: avoid lights shining directly into the camera, and avoid sudden, dramatic changes in lighting.
Also, don’t rely on your camera to record the sound of the event. It will probably do a fine job, but I recommend you have at least two other recordings to work from. If your church has a mixing-board for the PA, you might want to record separate tracks from its channels to ensure you can make a clean mix later for the video.