You light spontaneous portraits with available light. Sure, flashes, strobes, kinoflos, etc are great tools and really expand your visual vocabulary, but at the end of the day, nothing is as authentic and versatile as available light.

Start observing light. Find a cool light source - be it a back lit outdoor ad, a store display, etc. Look at how it lights people. Go with a friend, ask them to move around the light as you watch them. See how the light breaks, how it changes their features. Have a couple go-to moves - a direct ceiling light from above, a shot incorporating street lights, maybe a shot in the shade with a blue sky in the background, the specifics don't matter. Just don't stop experimenting, there's so much awesome mixed light in most city environments, it's a shame to overpower it with strobes just because that's the easy out.

If you absolutely must buy gear, get a Speedlite - a 580, 430, doesn't matter - and a cable. Being able to remove the light from the camera and movie it around your model is absolutely crucial. Learn to control ambient via shutter speed and flash exposure via aperture, there's a lot of great info on doing that on strobist.

And remember, there really are no correct answers, photography - especially non commercial photography - is all about expressing your vision. Have fun with it!