If you are doing portraits every week, bite the bullet and save up for strobes. If they are at the studio, then super light and portable is not a big concern
Older less glamorous set ups, by current day standard can be found, and are entirely workable. I waited and watched, and found what I feel was a deal. I bought a Speedotron 2400w pack, with three heads and a flight case for the heads, three caps, three 7" reflectors, and 2-11" reflectors for $1200. The down side of this rig is the pack is heavy. Smaller more portable ones are available. The stuff is built to last though. I later bought three used soft boxes ( 12x18, 24x30 and 36x48 ) with compatible speed rings for $300. Wescott. Good brand, lightly used.
If you are doing portraits, a lot of the time the look you are likely to want will vary from wide open to more serious amounts of depth of field. Sometimes I think you might be hinting at the 'new' photo florescent lamps.
Continuous lights will have you working wide open all of the time. Use strobes unless you are doing video as well. Video guys in it seriously usually hate hot lights as well, and have good CRI rated multi-tube floros that have a look of a soft box. Co-Flos is the industry term I believe.
Umbrellas have a nice look, but the light goes everywhere.
Soft boxes make great key (main) lights and smaller ones or strip lights - a longer and skinner soft box make nice hair/accent lights.
Fill light is fine with an umbrella. Usually a background light can be direct, but sometimes a shoot though diffusing white umbrella works well also.
High key shots can be pulled off with umbrellas.
I am not trying to scare you off. I started with hot lights, and stuck with them for a few years.
Then I moved to rigs with a bunch of hot shoe flashes on stands with umbrellas. Better than hot lights, but this was pre-radio sync, and the cords and batteries (pre NiMh) were a hassle as well. This is where I learned that umbrellas spilled everywhere, unless you spend all day hanging flags from the bunch of extra and tall light stands that you don't have at this stage of the game.
The heat of hot lights was not compatible with a nice sitting experience, and is even hard on still life stuff. The new cfl continuous source ones I don't see putting out enough lumens to cut it for portraits, but may work for product stuff,where shutter speeds can be stretched , and the LED ones are not quite bright enough yet, particularly when sometimes you are aiming for a point source effect.