I believe the best way to get HDR shots is by using the Av mode, so the depth of field is consistent across 2-3 images.
Should I be switching to manual mode and changing the shutter speed with a fixed aperture?
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You should use the Aperture priority mode, and keep the aperture the same for all shots, otherwise you could get some odd focus differences between shots. The only downside is if you have to go low on your shutter speed and pick up motion blur.
The easiest thing to do is use the camera settings that do multiple shots (usually -2,0, +2) in rapid succession, in which case you don't need to worry much as the shots go off in seconds.
If you to use more images or a wider range of exposures than your camera's exposure bracketing feature (if it has this feature) provides, then switching to manual mode is the best bet. In addition to keeping a constant aperture I'd also recommend switching off automatic white balance in oder to keep the color consistent.
For HDR work I would use raw, if at all possible. JPEG compression loses information, like color information from dark areas, which means your HDR software has more inaccurate data to work with. This also solves the white balance issue, because if the raw sensor data is indeed raw sensor data, like it often is, white balance is not adjusted at all.
And yes, like you said, I would primarily bracket exposure time and keep ISO and aperture constant. Aperture constant to have the same scene, just with different amounts of light, ISO constant because high ISO values soon negate the benefit of exposing longer.