Welcome to the forum.
When deciding what type of lights you want to buy, the first question is if you want continuous lights (lamps) or flash/strobe.
Continuous lights could be any type of bulb/lamp etc. You could use tungsten, halogen, florescent, LED etc. You could get something very simple or you could get really expensive professional lights. The most useful thing would be if you could attach light modifiers light umbrellas or softboxes etc. As an example,
HERE is something simple and cheap.
The benefits of continuous lights is that they are usually simple and can be the cheapest option. They give you 'what you see is what you get' type lighting. You see the lighting as you are setting it up, then you just take the photo...nice and simple.
On the downside is that some types of continuous lights get hot, which can make it uncomfortable for models and heat sensitive products.
Also, with continuous lights, your shutter speed is a factor in getting your exposure. You probably don't want blurry photos so you would need a faster shutter speed, which may mean higher ISO etc. Especially important when shooting people because they move too. With product, you can simply use a tripod, which allow you to use any shutter speed and still get sharp photos.
So, most photographers do prefer flash/strobe because they take shutter speed out of the equation. Allowing for sharp photos, especially when shooting moving models.
So when deciding on continuous vs flash/strobe, the question to ask is whether or not you will be shooting people. If you are, then I suggest going with strobes (or very powerful continuous). If you are just shooting products, then low power continuous lights are OK, just make sure you have a good tripod.
If you do go with continuous, the key is to make sure you get all the same type of lights. Mixing different types of lights will cause color temperature & white balance problems.
So if you decide that you want to go with flash/strobe lights, the question is whether you want 'flash' units (small units that typically run on AA batteries, or 'studio strobes' which typically run on AC power (but can be run on batteries).
You can use either for product photography.
Obviously, the benefits of flashes are portability. You can put them anywhere, take them anywhere etc.
The down side is that the good ones are really expensive ($500+) and they just aren't as powerful as studio strobes, especially once you start using modifiers.
Studio strobes are good because you plug them in and you don't have to worry about batteries. You can buy battery packs and take them on location anywhere, but the lights are big and the batteries aren't small etc.
Studio strobes are usually build to connect to all sorts of light modifiers and accessories. So if you plan on running a photography studio that can light just about anything...it will be handy if your lights can easily accept all sorts of accessories. Note that the attachments are not universal, typically you have to stick to the brand/style that you buy into.
The benefits of flash/strobe are that the light is a burst, therefore your shutter speed doesn't matter. You can get sharp photos without a fast shutter speed and without a tripod. They typically give you a lot of light, which allows you more freedom in placement/distance of lights and choice of aperture etc. Some even find that strobes are too powerful for some situations, even at the lowest power settings.
Also, with flash/strobe, the lights is not 'what you see is what you get'. Strobes often have a 'modelling light', but when you fire the lights and take the photo, the light is not really what you were seeing with your eyes. This means that you have to learn how to use them and usually take a lot of test shots as you learn.
It's actually pretty hard to know what will work best for you, until you actually start shooting. But how are you supposed to start shooting before you have the lights? Tough question. Ideally, you could use someone else's lights and see what it's like....or maybe rent some.
As for how many, and what accessories....again, you won't know until you are actually doing it.
I think that it's good advice, when you are learning how to light, to start small with maybe one light an a reflector and learn what you can do with just one light. When you start adding lights right off the bat, it gets much harder to learn.
If you haven't yet, get the book 'Light: Science & Magic'. Is it pretty much the studio photographer's bible.
