Hawkeye is absolutely right: A single, off-camera flash is an excellent way to begin to explore portraiture. The only additional piece of gear that you will need that hasn't been mentioned yet, is that you will need some way to trigger the off-camera flash to fire when the camera's shutter is open. The 2 easiest and least expensive methods are a long (eg, 10-15 feet) flash cord, or an inexpensive optical trigger. The latter start around $15, and go up to well over $100 each. Here's
an example of an inexpensive optical trigger that will work quite well for indoor, close range work.
Attached immediately below is a test shot from a series of photos I took just a couple of days ago for friends who had just brought home a new baby from the hospital. All I used was a single off-camera flash and a relatively big (60") umbrella, with the new dad standing so close to the umbrella (and slightly front of the center line) that he was practically enveloped by it. As you can see, it can give beautiful, wrap-around light, yet still preserve male skin texture. Changing from a silver lining to a white lining, is the way to go for pix of women, who typically don't want as much skin texture.
There are many good tutorials on single off-camera lighting technique available on the net, but here's one that I particularly liked because it shows the various looks one can achieve with this technique:
https://robertharringtonstudios.wordpress.com/
That being said, an even cheaper method is to simply hang a piece of white sheet in front of a window, and stand the subject close to it. This method can give equally beautiful lighting, but it usually isn't quite as bright as a strobe into an umbrella, so you have to crank up the ISO a bit. There also are many tutorials on this method that discuss the details. Here's one I liked:
http://www.picturesbymom.com/photography/how-to-take-a-beautiful-natural-window-light-portrait.html
To be honest, I haven't used the sheet-over-the-window technique in probably 20 years because it is so dependent on time-of-day, weather, availability of a window facing the right direction. If I go to some location to take a photo, I need to be able to generate a good image almost immediately, whatever time of the day it happens to be, without any waiting for the sun to be in the right part of the sky, and other fussing around. Using a flash completely eliminates the photographer's dependence on such factors.
HTH,
Tom M