andy,
Hi, and welcome to the NutzHouse! ;D
One thing I can tell you right off is, don't hook any of the output jacks's grounds together! That will almost guarantee some kind of ground loop problem somewhere down the line. That would not be a "good thing". Instead, do things in this order:
ONE: Keep all pickup/control/jack systems completely separate. Nothing touches nothing. Unless you have metal pickup covers (possibly they're humbuckers?), you shouldn't have any problems from just touching a pickup cover, so this should not be a source of hum. Be aware that if any two of the pickup covers are metallic, and if you should touch both of them at the same time, then you
will be completing a circuit, on the ground side, and you may be subjected to unsafe electrical conditions, meaning, the normal house-wiring voltage for your country.
TWO: If you should experience some hum or buzzing, you can try shielding the whole guitar, per the
"Quieting the Beast" article, and then hook that shielding to just one jack (thus to only one amp). That would help the remaining two pickups too, without introducing the aforementioned ground loop issue.
THREE: If you still suffer from hum/buzz issues, then you'll need to use the DI box method, noted above by ChrisK. But that requires 3 separate boxes, which aren't all that cheap when you need so many of 'em. However, doing so allows you to hook all three pups to the same shielding system, and then pipe the results to the DI boxes. Those boxes will restore the isolation between each the ground of each amplifier, and not incidentally, keep the guitar's shielding from actually becoming part of the overall ground system (that's one of the main functions of a DI box). While more expensive, it is the about the best way to assure noise-free operation and total signal separation for each amp.
When it comes to grounding the strings, you don't have a lot of options regarding the first two choices, you'll have to run the wire from under the Bridge to the ground pin on one of the output jacks. From that, I once again caution you (or anyone else in the same boat), if you touch the strings and a metallic pickup cover at the same time, you may be in jeopardy.
Grounding the strings under Choice #3 above is straightforward.
So much for grounding, let's deal with the switching issue.......
Normally, we speak of using a selector to combine pickups in various combinations for a single output, but your multiple-output rig precludes that, so it's a rather narrow set of possibilities for you. The only way you can get all three pickups to work at once, or any two of them, is to use individual on/off switches, one per pup. You
could use a rotary switch and get all 7 possible combos, but that's up to you. Me, I'd consider that overkill, and stick to the simple arrangement, but others here (ChrisK among them) will tell you otherwise. It'll boil down to what works best for you, that's all.
One other thing to think about. All though this is a bass guitar, the same factors apply as in a six-string rig, so I'll touch briefly on phasing between pups. In general, we do this right on the rig, with a single output to a single amp. (Meaning, to a single acoustical source of sound.) However, using multiple source points for our overall sound, we can get some interesting effects that don't show up when the phase switching is done electrically, on the guitar. This is a psycho-acoustic phenomenon, and deserves to be investigated by tone-moguls.
It only means that two of the pups need to have a DP3T (center off)* switch, which is readily available.
HTH
sumgai
* This switch is also sometimes labeled (erroneously) as a DPDT (center off) switch. Pay the label no mind - if it has a center-off position, then there are three throws, plain and simple.