The first thing I suppose I should talk about is the auto bracketing, on my Nikon camera it has a BKT button that I have to hold down, spin the wheal, and dial in the setting I want. If you have a different model Nikon or a Canon there is a 90% chance you camera will have an auto bracketing mode, just read the manual to find out how to access it. My camera only allows me to do 3 frame bracketing so thats what I do, I shoot it at 2 stop increments, if I have it on a tripod I review the images to see if I have all the exposures I need if not I go and manually shoot the exposures.
In the camera there is a setting that allows you to choose the order in which the images are put on the card, I choose the Under MTR Over setting this puts the darkest images first followed by the metered image and then the brightest. For me this is the best way because when I am back at the computer it just makes them easier to review in some kind of order.
Continuous high mode is the mode I bracket in 90% of the time. Its the fastest way to take your exposures because you just hold down the shutter and take advantage of your cameras frame rate. I do this because when I am hand holding I want to get through them quickly to eliminate as much movement as I can, when you come to make a tone mapped image this reduces the ghosting and saves you time sat at the computer editing.
When I am on a tripod continuous high is not the best tool for the job, if you are holding on to the camera this may cause some movement that you can see when your exposures get up to 1 second or longer. Because of this when i am on a tripod i don't want to be touching the camera, the one way is to use a release cable but i very rarely take one on a day trip out, so to get around this i turn on the bracketing and put the camera in to timer mode after i have pushed the shutter the timer goes off and the camera takes the 3 shots. You can customise the timer mode in the menu, typically i use a 2 second timer with a 0.5 second gap between shots, if my tripod is high i may put the timer to 5sec to allow time for the camera to stop any movement before the camera takes the shot.
OK at the top of the post I said I bracket everything that isn't strictly true but I do bracket most things like out on a walk or shooting stationary subjects. The time I don't bracket is when I am shooting a portrait, I don't want the shot when the subject gives me that look to be 2 stops over or under exposed. Another time is doing action shots I don't want that decisive moment to be too dark or too bright.
Generally bracketing is a very handy tool and don't worry if your camera dose not have auto bracketing, it is simple enough to adjust your shutter speed and manually set up the brackets with honestly a lot more freedom and control of your brackets.