As Jerry mentioned, you could probably do well enough with a set of 'hot shoe' type flash units, umbrellas and light stands. You can learn a lot more about this type of set up by reading
THIS.
There are pros and cons to this method. One con is that they don't put out as much power as typical 'studio' strobes and the recycle rate is slower, which can be a pain in but when shooting kids etc.
Since you will be shooting in your studio, then AC powered 'studio' strobes would probably be a good investment. There are several 'entry level' brands of studio lights, one of the more popular is Alienbee.
HERE is a 4 light kit, but you can pick and choose what you want rather than buy a kit.
You could also look into other brands like
Bowens,
Elinchrome etc.
A good set of studio strobes and accessories (light stands, light modifiers etc) is going to be more than just a few hundred dollars...but if you are serious about this, then you will probably end up getting this type of equipment anyway...so it would probably be cheaper to just get it up front rather than waste money on equipment that you will be unsatisfied with fairly soon.
Lighting for photography is partially about equipment but also a lot about creativity. You can do a lot of neat things with home made equipment and some ingenuity. Some people recommend starting with only one light, so that you are forced to learn the basics, then add more lights and equipment as you need them.