FWIW, Michael and I are of similar minds when it comes to lighting and a preference for using strobes that can be modified/controlled. There are dozens of choices, and the best choice for you depends strongly on your personal objectives. Your basic question, however, is one that faces virtually every photographer interested in making portraits.
The universal problem we all face, however, is not knowing enough about what we really want before we go looking for it. Thus, before delving into actual product choices, you may find doing a bit of reading to be helpful. Although not specific to portraiture, the best book I've found on general lighting is Hunter and Fuqua's book, "Light - Science and Magic". It will help you to understand the underlying nature of lighting, and how different surfaces (subjects) require different types of lighting. From there, a book on portrait lighting will help focus your personal choices. There is no single solution that meets everyone's objectives.
Additionally, there are numerous previous threads on this general subject here on APUG. Reading those threads may help narrow your choices, as well.
Although portrait lighting can be accomplished on a tight budget, I think you'll find that doing so is frought with compromises. You can, for example, start experimenting with two or three $10 shop lights from the hardware store. Or, you could opt for inexpensive strobes ($100 or so, each), or even multiple battery-operated electronic flash units (e.g. several Vivitar 283s linked together with sync cords). All of these choices have trade-offs in the nature of the light they output, the quantity of output (i.e. the exposure values associated), and how convenient they are to use - both for you and your subjects.
One important point with all of these choices - they will all work with virtually any camera capable of manual exposure control - as long as the camera or lens has a sync connection. The wrinkle is that you'll need to have a flash-capable light meter, so you can set up the multiple lights properly and determine the resulting exposure. (The alternative is to calculate exposure based on guide numbers, which gets complicated with multiple lights.)