Should I get a digital SLR video camera, or a camcorder, or something else?
It depends on your planned usage. Canon and I think Nikon DSLRs are still limited to recording up to 4GiB of footage, and then they stop. This works out to roughly 15-20 minutes of footage per shot, depending on bitrate (which will be influences by shooting modes and the amount of motion in a shot). Some mirrorless cameras, including the Panasonic Lumix series and the Sony Alpha series, feature shooting modes that can record for as long as they have power. So if you're going to have a dedicated camera operator, or plan on primarily shooting everything in quick, small shots, a DSLR might work out, otherwise look at cameras that use AVCHD, XAVC, or some similar system to overcome clip length.
DSLRs and mirrorless cameras also typically do not have servo-controlled zoom mechanisms, so any zoom changes will need to be done by hand, and if you're planning on shooting any of these zooms they may look unsteady if you don't use some kind of additional hardware to smooth things out. On the other hand, that means you can do things like snap-zooms if you so wish.
The other thing of concern is rolling shutter. DSLRs use a CMOS sensor which is better for still photography, while camcorders use CCDs, which are better for video. The problem with CMOS sensors is that while they are cheaper than CCDs, they don't have what's called a "global shutter," and each imaging line is scanned individually, which produces a rolling shutter effect (AKA: Jello-vision) when there is a lot of horizontal movement. Most modern DSLRs have this tamped down to a reasonable level, but it's still something to look out for when choosing a camera.
Should I try or try not to couple my audio and video?
You should keep the two separate, if possible, and not mix in the field. Or if you do mix in the field make sure you have an isolated recording. Mixing in post will give you the maximum amount of flexibility and will reduce the amount of opportunities for things to be screwed up on accident, which would necessitate a reshoot.
If you don't want to synch by hand you could look into Red Giant's PluralEyes and have it do the synching for you, just make sure you're recording some kind of audio on the camera.
Do I really need lighting equipment? I see people producing tolerable youtube vlogs with what seems to be minimal equipment.
Generally speaking, yes, you should have some kind of lighting setup. The more light you can throw on a subject the better, and you should definitely have one if you are going to be shooting in different environments so you can make sure you always have sufficient light.
For a typical interview a three-point lighting setup is recommended, and thankfully with LEDs becoming cheaper you can often find some small, cheap, light lighting kits that don't produce ten tons of heat. You can also experiment with different kinds of cheap lighting to suit your needs, like getting a cheap utility light from the hardware store and throwing a bunch of diffusion on it to soften the harshness.
I would also recommend getting some sort of bounce board to help better control your lighting. You can use colored boards to add a subtle "look" to your video (using a gold bounce board is common to help give a shot a warmer look), to soften shadows, or as a flag to control where your light is being thrown to reduce glare, lens flares, and reflections. There are nice ones that are very portable, alternatively get some cheap foam core from the local office supply shop.
Are there certain cameras that are betting with not great internal lighting than others?
Yes. A camera's light sensitivity is, generally speaking, governed by the size of its imaging sensor. The bigger the sensor the more light can be focused on it. Professional camcorders use sensors that are two thirds of an inch (measured diagonally), while a lot of consumer camcorders use sensors that are only one third of an inch. In DSLRs and mirrorless cameras the size can be almost anything. It could be as small as 0.85 inches, as in Micro Four Thirds, all the way up to just over an inch in the case of Canon APC-S sized sensors.
Also different sensor manufacturers may do better or worse in terms of performance at the same light level. For example the low light performance of the Sony A7s II is way beyond anything Panasonic can deliver at that size.
What is the best way to record the music that the instructors will be demoing to?
Add it in post. This will give you maximum flexibility to keep the levels constant. You can use recording from the onboard camera mic(s) to synchronize it to the music. I've never tested it, but PluralEyes may actually be able to do that for you too.
What am I overlooking? This world is new to me, but I'm excited about entering it.
A good tripod. If you plan on doing any tilting or panning or any of that jazz you want to make sure you get a tripod with a fluid head. You may also want to consider some kind of shock resistance system if you're going to be shooting on floors that have a lot of flex to them.
If you anticipate a lot of moving around you should consider maybe a monopod (monopods can also be used as a jib if you're willing to risk it), or, if you plan to be shooting while you move the camera, maybe a Steadicam rig or some kind of gimbal. There's a ton of guides out there on how to make your own Steadicam-like system, and most of the time they're pretty inexpensive to build.
You should also consider lenses if you're buying a camera with interchangeable lenses. The lenses included in basic kits aren't bad, but you may find them somewhat limiting, such as a limited "zoom" range, or it doesn't open up wide enough. You may want to consider adapting different kinds of lenses to it, but keep in mind lenses with electronically controlled focus and aperture may require some kind of powered adapter (one brand is SpeedBooster by Metabones), and those can get kind of expensive. Sometimes you can get away with just using a non-electrified adapters can also be used, and are very cheap, but you will not get autofocus, and you might not have control over the aperture.
When shooting it's strongly recommended not to use auto-focus or auto-exposure, or for that matter any other automatic controls. The results are almost never as good as doing it yourself, and the systems are easily bamboozled. Before you shoot, and after every time your lighting changes, you should re-white balance your camera using a clean white sheet of paper being held exactly where the talent will be standing.
Don't get caught up in the whole 4K/UHD thing. Distribution of 4K isn't quite "there" yet, and there isn't a ton of demand for it outside of enthusiast circles. It does give you some flexibility in post production, however, to punch in closer or for image stabilization, plus additional color detail when downscaling, but I wouldn't rule out a camera because it doesn't shoot 4K/UHD.
Focus a lot on sound. Microphones will pick up all sorts of extraneous noises you don't hear, like the sound of HVAC systems circulating air, or the hum of a refrigerator's compressor. You should have your sound engineer listen for these things and make sure they speak up if there's a problem with the audio. Don't be afraid to do things like post signs outside telling people you're shooting, and unplugging appliances (a fridge will keep cool for however long you're shooting).
- Always shoot some test footage and have your talent test your microphones.
- Always arrive early
- Never stop your audio recorder if the camera is still running
- Always run the camera a little longer than you think you need to, you can trim the slop in edit.
- Make sure your talent stops and holds position when they're done doing whatever it is they're doing. This gives you more flexibility in post.
- Use your audio meters to ensure your audio is mixed to around the same level, but don't "ride the levels" and be too fine grained
- Don't go too extreme with your color grading