Product photography with still images is typically done with strobe (flash) lighting. And most of the expertise here for product photography will probably assume flash rather than continuous lighting. If you need help with video aspects, I'd recommend asking a separate, more focused question on video.SE.
The number and type of lights that will best suit your situation depends on a large number of factors, the least of which is what camera you're using. While a cellphone or P&S camera may limit your aperture and iso settings, this type of photography tends to use smaller apertures and lower ISO settings anyway for more DoF and crispness, so the amount of light required from your illumination is going to be the same whether you use a P&S or a dSLR or a mirrorless camera. My only recommendation for a camera is to get one with a flash hotshoe so you'll have an easy way to use remote flash triggering systems.
The size of the product you're shooting, how much money you have to spend, and how you visualize and plan to set up the shot will be the biggest factors. There is no set answer that's going to work for everybody in all situations.
A light tent and small clamp lights is a good way to get started, but it's an extremely limited setup and over time, you may want to evolve to using multiple off-camera lights with modifiers just so you have more working room--a light tent gets awfully cramped and can limit your light, subject, and background placement possibilities in a way that makes it impossible to get the shot you wanted.
If your setups are small (say, food shooting), you may be able to get away with LED panels, and those could also be useful for video as a continuous light source, but they do tend to be lower-powered. Flashes are the usual photographer's tool of choice because you can get a lot of light out of them for very little heat/power/cost, and you can scale up to studio strobes vs. battery flashes if you need even more power. With continuous lighting sources, the cheap ones tend to be very low powered, and the adequately powered ones tend to be very expensive or very hot.
Lighting is a very deep subject in and of itself, and I'd recommend doing some reading to learn the subject. The standard college textbook for this is Light: Science & Magic. And the most popular website today to learn about off-camera lighting is David Hobby's The Strobist, although there are myriads more that have sprung up since that site started.