Due to the noise and unsightliness of the pump, wife is wondering if we can move it outside...
The primary factor in the relocation of your well pump will be the climate you live in. If you are in southern Florida and it never freezes, then things may likely work out well for you (no pun intended).
If you are in an area that gets freezing temperatures from time to time, things may not work and you could actually do damage to the pump, tank and other components if they were to freeze.
Sometimes there is a little "pump house" which is insulated, and also heated, which helps deal with climate issues, but heating a small pump house can be an additional cost, which one may find unacceptable.
One lower cost thing you could consider, is making an enclosure to cover the indoor components. Should you decide to do so, you may wish to make that enclosure readily knock down, so that you can gain access to the components without a structure around them limiting accessibility. You will also want to consider a double wall, or staggered stud construction to isolate noise from the pump. You will still have some noise. Better rubber isolation feet may help.
If your well setup is getting near the end of it's life, you might plan transitioning to a submersible pump.
A surface mounted pump can always be moved, yes they are noisy, and inefficient. This last year I changed my jet pump out to a submersible. Happy with the change all lower power bill.
If you have a jet there will be 2 lines to the well, basically you shorten these and extend the line to the pressure tank and extend the wiring. So yes it can be done.