Basically you have 2 choices, hotshoe flashes used off camera or studio flash.
If you already have at least 2 hotshoe flashes then that's your cheapest option (not the best but the cheapest) but if you don't have them then studio flash is cheaper.
The limitation of hotshoe flashes is that they don't have a modelling lamp, so you can't get any real ideal of what the light will be doing, so it's basically trial and error, and the tendancy for most people is to just stick umbrellas or softboxes into some kind of position that works, which doesn't exactly produce creative lighting. Studio flashes don't create that kind of problem and far more (and more effective) light shaping tools are available for them.
Studio backgrounds? A lot of people like blown white backgrounds, they're OK if you like tht particular style but they do need more equipment. Unless you have a lot of space your best best is probably the Hi-Lite, but whichever option you go for you'll need two flashes just to light the background, making the total number of flashes you need 3, but 4 would be much better.
If you don't want white backgrounds then you can either get a background support and paper, vinyl or muslin backgrounds and either light or not light the background to suit your needs - generally 1 light is needed for this. Or you can use whatever background already exists, one of the best photographers I know just uses a plain wall, although you wouldn't know it from his shots.
Finally, the best all round set up for most people starting out is probably 1 softbox, used as key light, and one umbrella for fill lighting. And, if you have a third head available for the subject, a honeycomb for use as a rimlight or hairlight.
