Tablet Solutions
For online office hours, I have access to an iMac with an attached 24" Cintiq Pro display from Wacom. Also connected to this computer is a relatively high-end webcam and a good microphone (you really don't want to be using the in-built microphone). This little studio is also equipped with good set of LED lights, which seems like overkill, but makes a big difference in the ability of your students to see you.
For video conferencing, I use Zoom. I am sure that there is other video conferencing out there (Skype, for example, or perhaps Apple's FaceTime), but Zoom seems pretty solid to me. Zoom can be used for free, but in a somewhat limited capacity (for example, I think that you are limited to 45 minutes of streaming before you have to restart your conference). In addition to letting students see your face (or whatever else a camera might be pointed at), Zoom makes it relatively easy to share whatever is on your screen. My understanding is that meetings can be recorded via Zoom, but I have never used that capability.
I used to conduct most of my office hours by setting up a desktop with Notability. Notability is designed for note-taking, and does a pretty good job of it. The only real downside that I have encountered with Notability is that it produces very large .pdf files. In the last year, I have started using GoodNotes—it has basically the same functionality as Notability, but is a bit cheaper (now that Notability no longer offers a completely free version).
While I was not a part of writing the grant which paid for this studio, my understanding is that it took about \$10,000 to set up, and the subscription to Zoom probably costs the institution something as well, so such a system probably requires institutional support and/or a healthy grant, and is likely not useful for all use cases.
If the above system is out of your price range, it can be made cheaper. Get a cheaper webcam and microphone, use a cheaper computer, and get a smaller tablet display. You could also use a tablet with a stylus as a writing pad, routing the video output through a computer to the internets (this is a strategy that I have seen used in large lecture halls). However, at some point you are going to start running into disappointing stylus experiences, poor sound quality, etc. Indeed, you can broadcast directly from a tablet using Zoom, if that is what you want to do---I assume that other video conferencing software behaves similarly.
Alternatively, skip the tablet entirely, and get a couple of high-quality webcams. You will want one camera that shows your face, and a second set up on a tripod which can be used as a document camera. Get some good lights to illuminate what you are writing, and stream that. There are a couple of downsides to this solution: for example, it is an extra step if you want to upload written notes, and while you are writing your hand is going to be in the way. However, the biggest expense here is likely to be the computer, so it could be done this way much more cheaply.
Learning Glass
Another possibility is to use a Learning Glass setup. The Learning Glass is essentially a large piece of glass with lights embedded around the edges which shine into the glass and illuminate anything that is written on it with neon whiteboard markers. You stand on one side of the glass, and a camera is set up on the other side (the image is flipped horizontally so that you can write naturally on your side of the glass, but your students will see everything with the correct orientation---there is only a problem if you are overly attached to your students believing that you are right handed). Our department uses the Learning Glass to produce "micro-tutorials", which you can have a look at if you are interested.
I am not a huge fan of the Learning Glass. It takes up a lot of space (you need to setup a large sheet of glass, and have space both for lecturers and cameras), and it is quite expensive (on the order of \$12,000-15,000 for a studio setup, though they do sell a desktop version for about \$5000). You have to be careful about where you stand, as you can stand right behind something you have written, which will make it hard to read. Also, make sure that you have a black backdrop, and get used to wearing black (ideally). There is also an issue of resolution---you set up the camera, then have to perform for that camera. In contrast to what is done in Notability (or other similar software), you cannot zoom in on an important part of the screen in real time. Finally, and this is maybe a bit nitpicky, I find the neon markers hard to erase.
More cheaply, you could set up a camera in front of a whiteboard, then simply lecture as usual.
For streaming, something like Zoom is still a viable option.
