Edit: Ihasa posted while I was still writing, but I'll post this anyway...
As a fellow K10D owner I can relate to your frustration. It's a great camera, but since the sensitivity range tops at 1600, it's not ideal for indoors or anything fast paced. However there are a few things that can help you get the best possible results from your GX10. But first I'll see what the two pictures you've posted can tell me.
On the first one you're shooting at the maximum aperture of the kit lens, which is by no means a bad performer, but neither does it shine when used wide open. The more important thing is the shutter speed, which seems to be 1/15s. This is quite slow for anything that can move the tiniest bit. This can be seen from grandma's hand, which actually creates a cool effect to the photo. However your son moved his head just a little bit which causes it to blur. Indoors is a difficult shooting environment, and even when your eye says there's enough light, there's usually not too much. This can be overcome by two different ways.
Get yourself an external flash. Anything that has the capability to at least tilt and maybe swivel. This way you can bounce the flash from the seiling and your images will appear normal, not like the ones where you use the on-board flash directly. Also by starters you can reduce the comparable output of the flash from the flash menu in your GX10. This way more ambient light will affect the image and create a more natural look.
Another thing to do is get some faster lenses. A good one for indoors would be The DA35 2.4, which will give you more light and thus you can bump your shutter speed a bit resulting in sharper images. Another good/cheap alternative is the DA50 1.8, but it's a bit too long for indoor use. Or if you put out a little more cash you'll get a Tamron 17-50 2.8 which is a great performer given its price + you get the handiness of a zoom lens.
On the second picture it seems like the focus is behind your friend and the fish. Most likely the eye is a bit too small target for the autofocus system, ans it focused on the scene behind them. If you had chosen for example the face of your friend, given the 18mm focal length and 5.6 aperture, the fish would've been easily in focus too. The focus points are really quite large and you need to manually check that you've got focus on where you want it to be. That's what viewfinders are for
Had you had proper focus your GX10 would've beaten the iPhone by miles, however it has such a small sensor that practically everything is in focus on these distances and he doesn't have to worry these things...
The GX10 is a great camera as I stated, but you need to know how to get past it's slight flaws. Getting good lenses is the way to go IMO, and once you get enough cash you can upgrade to K5 which will bring you improved high ISO performance + a lot of other things. If my explanation remained unclear to you, please don't hesitate to ask more.