For best results you should set your camera to full manual. Imagine the batch of images you’re about to take is a single image. Any slight difference will damage the stitching process and final result. Here's a checklist for all the settings you should lock:
White Balance (especially important if shooting .jpg images)
'White Balance' can be left on auto if shooting RAW files
Be careful when setting the exposure (see next paragraph)
Try not to knock the focus ring when rotating the camera throughout the panorama
You can also shoot video for this. Same rules apply for settings (log profile recommended for better dynamic range)
It's important to plan your exposure Since you'll only be looking at a small portion of the final image at any given moment. I try to frame the focal point and expose for the subject in that area as best I can. If I can also fit a tricky part of the exposure in to that frame (like the background highlights, below) to also keep in mind that will help. Shooting RAW will give you much better flexibility of course, but this is no substitute for setting a good exposure.
Tripod Use (Normal)
If we're talking about a normal tripod head (not a panoramic head) then you mostly should not use one. This is because any standard tripod head (ball, pistol grip, pan & tilt etc.) will rotate the camera & lens in all the wrong axis. You're much better off just guessing the rotation. The caveat here is longer lenses. The incorrect rotation is not so problematic on lenses longer than 200mm and there you will appreciate the support much more.
Correct Rotation Axis' - (Panoramic Head)
Adding a panoramic head to your tripod goes from being a guaranteed failure to assuring perfect rotation. Set up properly, this will avoid parallax errors (with static objects) and increase your chances of a reliable, artifacts free stitch.
So, do you need one of these? No, I don't use one and this is why... I like to travel light so I don't like carrying around the extra weight of a tripod and panoramic head (also I try to keep my camera/lens weight under 1.3kg). I also don't like dealing with the extra time it takes to set up a tripod and then move it around. I find guessing the rotation point of the camera/lens almost always good enough. Most stitching errors I get are easy enough to fix in Photoshop. It is however very useful to see this setup and trying to emulate this movement during your stitch.
If you really want to use a tripod with a panoramic head and don't mind the carrying the extra weight, then go for it. It will improve your stitching / frame alignment. If you have a tree blowing in the wind or a person in your scene then a panoramic head could slow you down and increase the chance of errors.
As well as the rotation of your camera during shooting your batch of images (see tripod section above) there's also direction to consider. There are many ways to do this. Some use a spiral, some a zig-zag. I tend to favour a horizontal variant of the latter (shown below). A spiral method is probably the best for shooting people (starting with the face) because you'll shoot the face straight after focusing (before drifting) and if the subject moves, blinks and/or drifts you're more likely to avoid that in the early stages of the batch.
I recommend aiming for 50%. Moving the camera half way along the previous frame for each movement. It might seem like overkill, but remember than any slight slip could reduce this amount and the stitching software needs matching elements to align the images later on. If you get really good at this you could aim for 1/3, but this will result in more failures.
There is no hard rule. As you can see from the above example I prefer the portrait orientation. This is probably due to the horizontal movement and trying to perceive the best overlap, but I’m not quite sure. If this feels really uncomfortable to you then just switch to the alternative. Whatever feels right for you is the best way to start, just keep in mind that the other method might end up stitching more accurately for some weird reason.