Continuous lights are HOT, if they provide the light needed for a decent shutter speed. I discourage them unless you are shooting inanimate objects. (I have three 750W variable output backs for my softboxes, but they rarely get used.)
You can go the battery flash route; any unit with manually-controlled output levels can be a fair starter...such as the vivitar 285's (or many others)...which can be picked up for a few 10's of dollars. I started with vivitar 283's. with stands, brackets, clamps, optical slaves, umbrellas and even a modest softbox and your time and enginuity(sp?) you can control the lighting and do some fine work. Since you have a modern digital camera, you may want to go with a camera-compatible flash such as the Canon or Sigma or others... for the canonflash communications when you shoot that style, and the manual controls when you want precise control.
You can patiently search the yard and estate sales as well as local camera shops or eBay for buys ...to get you started. I picked up a 2-head Novatron kit with stands and umbrellas and case at a yard sale. The kit is fine...for many applications (Like volunteer work - Pet Pics with Santa for the Humane Society)... I picked up a pair of nice German-engineered studio strobes at full retail when they finally got off back-order... I knew I wanted them because I already had the softboxes to fit them (from another lucky purchase). Just today I browsed the remains of an estate sale, and picked up a pair of stands, tilt-umbrella clamps, and WORKING "Spiratone Flash Bank AC Studio Flash" 135ws heads... for $20 !!! Not sure when/where/how I'll use them, but they work, have variable output and optical slaves. Yes, as an engineer I tend to have double or triple redundency in bodies, lens coverage, battery as well as AC flash units.
Suggestion - get ONE flash unit, stand, umbrella or softbox, and a LARGE white reflective surface (white foamcore board) and learn to see how the light wraps the contours of the subject(s) before adding a 2nd, 3rd, or more. TOO many variables can quickly muddle your efforts.
Chuck G.'s instructional materials are worth the read. A pair of the fun-foam flash modifiers are in my camera bag too. ;^)
Enjoy