IN beds, apply mulch (wood or bark chips). Rake it around every so often and the moss will not be able to establish itself on the mulch.
IN the lawn, try to keep the site as dry as possible (cut back on watering, reroute downspouts, improve surface drainage, etc), raise the mowing height (so the grass will shade out the moss), and if the site is shaded by trees, see if you can remove a few branches and let more sunlight reach the grass.
There is moss killer that you can buy to get rid of the moss in the short term. It's iron sulfate, or something similar, and shouldn't pose danger to pets. But the long term solution is make the changes detailed above.