You can get started with a few things:
1. A basic 4-section ~7 foot light stand -
this Manfrotto Nano stand is just tall enough and very compact. The larger 4-section stands that go to 7.5 feet might be better in the long run.
2. a simple double fold white "shoot thru" umbrella - I've
used these Westscott for years3. an umbrella stand adapter,
this one seems reasonableYou can buy all of these things much more cheaply on Amazon, and those will all work fine too. You can also get inexpensive kits, look for Cowboy Studio and the like.
I like the shoot through umbrella - easy and fast to set up, nice light quality. You don't get a huge amount of control over the light, but it's very forgiving - just put it a few feet away at a 45 degree angle, just above the subject's head and pointed slightly down and slightly in front of the subject (aim for the tip of the nose.) Then play with TTL and Manual flash modes, and vary your ambient light exposure and see what the background does. Move the flash around and see what that does - directly over the camera with a big umbrella gives an interesting look, as does from a 90 degree angle or even more to the side for a profile shot.
When you really want more control over the "spill" of the light (what it's hitting that isn't your subject), then you can get an Apollo Orb and its associated fabric grid for a very reasonable price. It'll mount to the same stand and umbrella adapter, but provides far more control (and far less forgiveness
) I've been using one lately with my radio controlled speedlights, and it's very nice for a one person portrait.