Montoya,
It all depends on how much money you want to invest, what kind of support personnel you have (just you, or you plus others), and what kind of quality you want out of the final product. I've seen good quality videos out of some phones, but if you are looking for image stabilization, HD or 4k video (or higher), etc you are going to have to spend some money.
Some people here have already suggested some good ideas. The Cannon and Nikon digital SLR cameras for instance are great from what I have heard. (The Nikon can do up to 60 fps, the Cannon 30 in 1080p) Otherwise, a nice VHS or digital video camera, GoPro, etc would do well.
Shotgun microphones are nice as are lapel mikes. If you are looking to do multiple microphones, a good mixer will also be needed unless you are doing multi track recordings and will mix later.
So, depending on what you want to spend, you can go with your phone for minimal expenditure, 100 bucks or so for a cheap digital video camera, to $2,000+ (maybe even $5,000+) for some DLSRs (oh, and you will need some lenses as well...figure 40-1,000 or more depending on the type and quality (bigger the lens...not just the zoom level...increases the costs)
With the budget numbers you gave above, If you have an iphone 6s, it gives 4k support already. Others are HD unless you have a much older one. There are some acceptable lenses in the $20-$40 range if you are wanting cheap. A lapel mike for $20, a shotgun mike for $40, and an image stabilizer/holder from $40-$150. I am sure there are similar things for android depending on the model of your phone. Heck, you can even get a microscope lens for 10-40 bucks if you are looking to add even more capability.
All this said, good cameras cost money, good lenses more, and if you are looking to do this in any kind of professional manner (aka, making money off it), the better the quality of stuff you purchase (lenses being even more important than the quality of the camera) the better the final product. Also remember, you get what you pay for. Going cheap now means throwing most of that money away later when you replace it.
