When you are talking lighting...there are two main categories...flash (strobe) and continuous lights (sometimes called Hot Lights).
There are pros and cons of each.
Continuous lights are easy, you set them up and what you see is what you shoot and what you get. Also, they are significantly less expensive than strobes.
On the down side, it takes A LOT of wattage from continuous lights to equal the power of a good strobe. This is important because with continuous lights you rely on the ISO, Aperture and Shutter Speed to make the exposure. The less power you have, the longer the shutter speed you will ultimately need. A long shutter is OK when the camera is on a tripod and you are shooting something that isn't moving...but if you are shooting people, a slow shutter speed will mean blurry images. Also, people don't like to have a bunch of hot lights on them for any length of time. Yes, the fluorescent lights aren't as hot...but it's still not fun.
Strobes on the other hand, fire very quickly...much faster than the shutter in the camera. This means that the shutter speed doesn't even matter much at all for flash photography. The burst of light is fast enough to freeze motion, so you can get sharp shots of people and don't really need a tripod. Having more lighting power also means you have more flexibility with your set up. If you don't have a lot of power, your light may need to be very close to the subject...which is fine for still life and not so good for portraits. There are some beginner strobe kits buy often they are under powered and would be limiting.
The down side of strobes is that you can't really see the result because it's so fast. A light/flash meter is often needed, although digital can make it possible for you to guess & test.
So I guess it depends on what you want to shoot. For still life or product shots, continuous lights would probably be OK. For shooting people, strobe is probably the better way to go.
Now...with strobes there are studio lights and 'hot shoe' flash units. Price wise, it's a toss up. Top end flash units are close to $500 a piece and you still need a way to trigger them off camera. Good Studio lights start at around $300-$400 a piece. The biggest difference is the power supply. Studio lights need to be plugged in and flash units run on batteries. Studio lights are also more powerful.
Flash units are pretty versatile and can also be used on camera when you want to...but the (Canon 580EX for example) at $500 (CDN) a piece, is really expensive. The Strobist has some great ideas but he has many years of experience. It takes a lot of time & effort to get results like that.