The first thing you’re probably thinking about is which camera to get. The short answer to that is, “whichever one you have with you”, but if you’re actually planning ahead for a video shoot, you will get better results from something a bit more purpose built.
Even though it looks like just a simple point and shoot, the Sony RX100 (Amazon.com link) is a great choice. It has a lot of video features packed into the small body. The optics and low light performance are good, which is important if you’re not going to use a lot of additional lighting.
A short word on lighting.
Having abundant light is one of the larger factors that affects how a video turns out. A lot can be done in production software, but lots of light always helps.
That being said, you don’t need to go out and buy a bunch of expensive lights if you plan properly. Natural light is always great, so if you can shoot your video outside or near a window, you should be in good shape.
If you can’t get enough natural light, turn on as much of your indoor lighting as you can. Just don’t put bright light sources behind your subject as it can “blow out” the frame and be distracting for the viewer.
Probably right behind the camera in order of importance is having a good tripod. It doesn’t need to be super fancy, but having a fluid head is a nice feature and this is the one I recommend: Slik Video Tripod
Finally, sound is a big deal in video, more than you would probably think. You can get away with a lot in the video frame if your sound is good. On that note, you will want to get a wireless lavalier (lav) mic. The most convenient ones will be battery powered on both the sending and receiving end. This makes it much easier to use than trying to find a power outlet for all the pieces and hooking everything up whenever you want to shoot a video.
Here’s a good, basic wireless lav setup: Wireless Lav Mic
One of the reasons I recommend the camera above is that it has a microphone input. Not all budget priced cameras do, so this is a critical feature.