It sounds like P mode on the mode selector dial might be a good place to start for you.
P mode is "automatic", the camera will select the aperture and shutter speed, and it sets them to maximize the sharpness of the lens mounted, the fastest shutter speed. The rear control wheel will change the aperture, and the camera will adjust the shutter speed appropriately.
If you've selected Auto ISO, P mode will start bumping up the ISO, but only as a last resort. Once the shutter speed gets to around 1/45 and the lens wide open, the camera will start bumping up the ISO as you turn the rear control wheel. This 1/45 of a second may be lens dependent, ie it might be 1/200 sec for a telephoto lens... Anyway, bumping up the ISO is the last thing P mode adjusts, just what you want. Try it and see!
Trust the K-x ISO performance - it really is quite excellent up to 1600/3200.
The P mode is highly automatic. After you get used to this and see what the camera does, move on to the other modes, like A mode to control the depth of field of focus, and T mode, to control shutter speed (like blurring a waterfall...)
I would also suggest using AF.S focus mode. This is very similar to point and shoots - push the shutter 1/2 way, and the camera will auto focus and lock the exposure/focus, and then a full push on the shutter takes the picture. In AF.S, the shutter will not fire until focus is locked. Try AF.S, center point only, or maybe the 5 point auto select. The 11 point and AF.C is more for moving objects like kids, birds, cars, etc. Your dog looked pretty immobile
BTW, shooting outside in snow, you need to over expose by 1-2 stops, otherwise the snow comes out gray, like in your shot. It's easy to dial in, regardless of what mode you are shooting - just push the +/-Av button and use the rear control wheel to dial in +1-2 stops.
Hope this, and the other posters suggestions, help.