Olympus e-PM2 - what am i doing wrong?

It seems like your issue is generally with shutter speeds being too slow for your liking. Keeping the camera in 'A' (Aperture Priority) or 'S' (Shutter Priority).

If you keep your camera in 'A,' you can make sure the aperture is as wide as it can be (for your lens, that would be 3.5 on the wide end and 5.6 on the long end). This means selecting the smallest number that's presented on screen - think of it as the smaller the number, the lower the light the camera can cope with. This would give the camera the best chance at selecting an ISO and shutter combination which are fast enough while keeping the image properly exposed. The other benefit here is that you should eliminate the risk of the camera auto-selecting small apertures (f/11 and higher), which can cause visible diffraction (sometimes even at screen-size) and potentially poorer AF performance (due to less light hitting the sensor).

If you keep your camera in 'S,' you can select shutter speeds that are fast enough to keep your subjects sharp. For stills, 1/(focal length x 2) should be fast enough, so if you're at wide angle, that would be 1/(14mm x 2) = 1/28 sec (closest on the camera is 1/30), and if you're at the long end, 1/(42mm x 2) = 1/84 sec (closest on camera is 1/80 or 1/100). For subject in motion (like bouncing kids), select a shutter speed at least at 1/200 sec - outdoors, I wouldn't hesitate to do 1/1000 sec or even faster. The biggest risk of being in 'S' is that you select a shutter speed that doesn't work - too slow, and you end up with a slew of blurry images (like I got at a recent wedding - 1/60 sec isn't fast enough to freeze motion on the dance floor); too slow and you risk either a) noisy images as the camera pumps up ISO to maintain exposure, or b) underexposed images as the camera can't get enough light.

Lastly, take some shots at ISO 1600, 3200, and 6400. Depending on your tolerance of noise in the image (graininess), it might be good to tell the camera to max out at one of these settings when ISO is set to auto. A max of 1600 is conservative but you'll nearly always come out with images that have acceptably low noise; a max of ISO 6400 gives you a better chance of freezing your subject (you can use a shutter speed 4x faster than ISO 1600), but you might not like the grainy or splotchy effect it has when you zoom in or crop.