OK, before you start talking about umbrellas...take a step back.
So you want some studio type lighting equipment?
The two choices are flash/strobe or continuous lighting. There are pros and cons to either choice, but for shooting people I would strongly recommend going with flash/strobe. You can still get studio lights that have a 'modeling light', which is a bulb that stays on, but that is only for seeing what you are doing, it's not what you use for taking the photo.
Now, if you are going with flash type lights, there are more choices to make. Firstly, you can go with actual studio style lights. They can be fairly big and need to be plugged in to a power outlet. Or you can go with hot-shoe style flash units. These are powered by AA batteries, so they are a lot more portable, but they lack the power of a good studio strobe light and they don't have a modeling light.
Once you make that choice, then you can start to think about light stands and light modifiers like umbrellas and softboxes. You will also need a way to trigger the lights. It could be a simple cord from the camera to one of the lights or could use some sort of remote triggering device. Studio lights usually have a built-in optical slave, meaning that if one goes off, the others do as well....so you only need to connect to one light. With flash units, you could would need something for each light.
A big difference between constant lighting and flash/strobe is that it's harder to find your exposure with flash because the camera can't measure the flash output. This is where a flash meter comes in very handy, although you can use the trial and error method when shooting digital and don't have a flash meter.
Here is something to look at for studio lighting.
AlienBees: Illuminating the Galaxy with Professional Photographic Lighting Equipment
Here is a 'strobist' (flash unit) kit
MPEX Strobist Kits
I'm sure you will have more questions and also note that this has been asked many times before, so don't forget to search the forum.
