For me, the biggest problem (capturing candid photos in/around the house) is with distracting elements that notoriously appear in the background. It seems the animals do the 'cutest' and most picture-worthy things when they are in close proximity to some large piece of furniture.
Combine that with the effect of an 'automatic' camera and its choice of when to use a flash and I often get shots with the corner of a table BLARING white in the edge of the frame. Many times even cropping is not enough, so I resort to image editing (mask the offending area and either burn or blur it).
Dodging and Burning are two techniques I had mastered back in Pre-Historic times with celluloid photography/printing. The digital counterpart is WONDERFUL because you can UNDO at will.
I think my biggest problem is I don't have a crazy high-end camera, so "action" shots always come out blurred. Even if it's not supposed to be an action shot... silly animals.
One useful technique, when trying to photograph any moving object (particularly an animal) is to move the camera with the object; tracking it in the frame.
Just imagine you are shooting video and squeeze the shutter at an opportune moment (or several times if appropriate).
You may get motion blur of the background, but that will only enhance the photo if the subject is in focus.
How in the world do you get your pet to look at the camera. It is such an uninteresting object for them to view so they just glance and you get .15 seconds to take a picture. I usually set my camera on the fastest shutter speed to try and capture anything.
I would say that my biggest challenge is trying to get great shots without being properly prepared. The great shots (especially hi res for printing) are rarely captured by accident or candidly. To get that great shot you need all the right conditions (gear, lighting, background, pose, etc) and that takes time and preparation.