Variations on a theme ... :-)

It partially depends on how "arty" or "utilitarian" you are trying to be.
A holiday 360 degree panorama of a mountainscape is probably aimed at getting the scene across.
An architectural view looking at a building with a street running away at an angle either side of centre my be trying to drop out backgrounds or ensure focus all the way up the building tower (or not) or some other effect.

In the latter case a general answer is hard as you need to customise to suit the specific aim.

In the case of a landscape panorama where "conveying the scene" is what matters most then some general comments may end up being relevant (hopefully)

Aperture smaller than larger such that out of focus is less of an issue or not an issue.
Consider manual focus (cf Matt's comment on hyperfocal distance) with focus "touching infinity" and close end of focus as close as that allows. f/22 will pretty much cover everything in most cases (maybe not the daisy at 1 metre, maybe), f/11 pretty much so, f/8 and larger will start to force decisions on what is in focus. Aperture affects this but in most panoramas you will be at relatively low focal length.

• That said - I shot a 180 degree + 64 frame maybe 100+ kilometre wide panorama of the NZ Southern Alps from a park near the coast in Christchurch. Camera in portrait mode, zoomed in until the mountains to the plains just didn't fill the frame at their visually tallest
  point, start at left end and move in just overlapping frames along whole mountain range. = 60+ images.

Shutter speed is a servant of the situation. As long as there is no dynamic content (animals people cars ...) that you want sharp then as slow as you need after all else is decided. this may be you main "free" variable. Note that clouds and traffic may fool you if not considered.

ISO sensitivity is quality related. Highest ISO that produces quality that suits you completely EXCEPT perhaps where you have a very bright scene and or may be using an ND filter for some effect.

Focus as above for aperture, plus - depends on scene. If there is nothing special or if everything is special you may need to focus frame by frame. If not, then a depth of field that includes the whole scene allows freedom to manually set and forget.

A contrived worst case frame by frame focus case may be standing in the middle of the circle of stones at Stone Henge (should you be allowed)(which you are not) and taking a 360 degree panorama where you wanted the stones in crisp focus but an absolute minimum, depth of field of backdrop beyond. (That would be an awesome photo!). Due to the uneven spacing and sizes and distance to various stones you'd need to focus each frame to suit the content.

White balance I don't recall ever having changed this during a landscape panorama BUT if you were eg shooting towards a sunset towrds the West and evening shadow to the East then a change of WB may well be in order. Auto WB may suit and may not depending on scene and camera.

Exposure - most tricky last - in my experience at least. I've found exposure the parameter I have to be most careful with. To some extent it depends on your stitching software and also on what effect you want. If you want a sunset to deep dusk contrast then changing the exposure to damp down the sunset and bring up the shadows may work exactly against the desired effect. If you have a scene where dynamic range exceeds camera capability (which is often the case and certainly so i the sunset 360 degree example) then a degree of adjustment will be required. This can be just 'autoexposure" but you need to trust your sensors severely. In days of yore (digital but pre-live-view) I would probably have trusted the wide mode light level sensing or done some test shots at key points, set to manual and autoadjusted where required along the way.
With live view (and this will be heresy to some) I am comfortable with using autoexposure per frame, previewing and then adjusting with over-ride (rear thumbwheel) where needed. eg in the example sunset 360 I'd probably drop exposure up to 2 or even 3 stops when shooting towards the sun and maybe raise it one or 2 stops at the extreme dusk portion with progression between these as I go - depending as above on where-ever "even illumination" or "looks like real life" is wanted.

Panorama stitching software makes a difference. I use free and amazingly capable [Autostitch] with a strong tendency towards record-of-journey panoramas rather than anything arty.

On the fly, hand held:

When travelling I will often take informal panoramas of a scene to get an overall feel. this is often done in shortest possible time with hope of absolute perfection a distant dream.
Quick spin around for anything obvious that will cause problems.
Decide what focal length is needed to allow optically largest target frame to look right. (eg a local hill or tree that you want to include will be taller than a distant one .
Set to autoexposure, aperture priority (my default). Aperture as small as reasonable. ISO as high as reasonable. Auto WB usually.
Photograph my hand in closeup :-) (Later unmissable marker of sequence start.

. Frame top left at height to be used as upper bound.
. Half pressure - check liveview exposure level. Adjust if needed. . Either manual focus already set or (usually) point at desired focus point with centre spot focus half pressure, recompose with top left correct.
. Take photo.
. Hold steady - check new target in top right which will be comfortably included in next frame with desired amount of overlap. (This is possibly done as soon as camera is set to new view at start of sequence).

Repeat until finished.

Take photo of my foot in closeup :-) - unmissable sequence end.

The above per-frame procedure may sound cumbersome at first glance but becomes essentially move-check exposure appearance-focus-recompose-shoot-move- ... at maybe a second or two per frame when there is nothing especially special in view.

The above gives me reasonably consistent exposure overall, focust is correct for whatever I want in focus and - two major features - overlap to the extent I have decided I want and a relatively level top line. It is VERY easy to take handheld informal panoramas where the horizon line is an up and down mess so that if you want to cut a rectangle out of it later you lose much material. By formallt siting on something new to set your new horizon and end point to BEFORE you move the frame you get a usefully consistent horizon line. The above is of course not as good as using a tripod and xxx degrees per shot, but works very well for very quick on the fly panoramas.

Revolving restaurant:

Set camera to 30s time lapse.
Place on table pointing out window.
Fiddle with serviette support until view is correct.
Zoom to suit.
East lunch.
Ensure meal is long enough for at least 360 degree sweep.
Ensure frames overlap very significantly (see below).

With enough overlap frames with window bars in may be discarded. Otherwise some cropping needed before stitching.

Set white balance by illuminating white surface with light through window with restaurant light shielded. This allows glass white skew to be removed.