I went back over this thread, and I'd like to chime in again.
First, it makes a HUGE difference whether your interest is formal portraits or more casual portraits and candids. There are exceptions, but formal portraiture often involves much more equipment.
Second, my advice is to keep it simple and to add additional techniques and equipment slowly, as you need to. For outdoor work, the key is learning how to use available light. You don't need additional equipment for that. For indoor work, you can sometimes use available light, as Manfred's example shows. However, most often, that only works if people are willing to sit still, and it only works if you have a good source of ambient light. If you want indoor candids, you will need artificial light most of the time. The key there is what Manfred said: avoid direct flash, because it usually yields ugly results--shiny faces, red-eye, and harsh shadows, among other things. So to do that well, the minimum step up--and I would stick with the minimum--is a flash that can bounce. Some off brands are quite cheap, but IMHO, it is very important to get one that works in automatic (TTL) mode with your camera. That allows the camera to control the amount of flash and makes candid work both easier and much faster.
With apologies to the people who have seen it many times before, I'll post a photo of what I generally use for candids. Sometimes I use a longer lens, but I almost never use any equipment other than the camera, the flash, the bounce card (the big white card), and a diffuser. The card and diffuser are cheap. I can't show you photos examples of the many candids of kids I have taken with this because I never post photos of kids publicly without permission of the parents, but it works very well most of the time.
![Starting taking Portrait Shots](https://photos.smugmug.com/photos/i-rMddWDz/0/d4c98ca4/X3/i-rMddWDz-X3.jpg)