First things first: the planet's magnetic field will not help you. A typical value for the Earth's geomagnetic field on the surface is ~30 microteslas, whereas a standard kitchen magnet is in the neighborhood of 5 milliteslas (= 5000 microteslas), or about 165x as strong. And kitchen magnets are not going to keep an island in the sky very long. In all probability you'd have to change the composition of the planet so much that it wouldn't really be Earthlike any more.
You could posit that the crust of your planet contains permanent magnets and/or superconducting magnetic material in abundance, which repulses bits of crust that are the wrong polarity: hence the sky islands. (This is how the unobtainium works in Avatar and Civilization: Beyond Earth; I'm sure there are others.) Technically speaking this probably wouldn't actually work, but it's near enough for fiction.
Permanent magnets in the crust in particular should pose less danger to computers and electronics in general. To oversimplify, electromagnetic theory says that changing electric fields cause magnetic fields, and moving magnetic fields cause electric fields. As long as they remain relatively stable, the effects should be minimal. If one of your islands starts moving around, or worse, crashes, that could cause a major electromagnetic incident.
As far as I'm aware, these sorts of magnetic field strengths wouldn't be directly harmful to life, and they wouldn't have an effect on most weather. (Having a giant, presumably ungrounded bundle of matter floating in midair might do something odd to thunderstorms, though.)
According to this paper, you can actually levitate live animals with a field strength of about 16 teslas, and it's perfectly safe.
Of course this represents the summation of microscopic currents localized in atoms, not the bulk transport of charge, so the living creatures were not electrocuted. Indeed, they emerged from their ordeal in the solenoid without suffering any noticeable biological effects—see also Schenck (1992) and Kanal (1996).
This is using a strong solenoid magnet on the one hand, and the fairly weak induced magnetism of the water in the frog's tissue on the other hand. It suggests that biology won't be a major source of problems.