If you shoot indoor photos, then you need a tripod and some external flashes/strobes. Strobes will cycle much faster and has more consistent exposure than the hot-shoe flashes. Add a softbox to your strobes and a backdrop will make your puppy photos look professional. This can easily be done without much expertise.
I don't take dog photos. But here is a photo of my 1 year old who moves just as much as any puppies and dogs. The minute you put him down, he will start getting up and walking away.
http://i165.photobucket.com/albums/u45/hjfok/Kids%202010/IMG_7827.jpg
I set up 2 strobes (one with softbox and one with an umbrella), a camera on a tripod and a simple paper backdrop. I have a light meter that comes with the remote trigger of the strobes, just press the button for the test flash, the meter will give you the optimal settings for the camera. Then you do a test shot with the settings to see whether you like the exposure, adjust if you don't. Then put your subject in front of the camera and snap the shot. It is quite simple and has consistent results shot after shot, don't have to deal with the TTL variations of the hot-shoe mount flashes.
I have a few Alien Bees strobes, they have reasonable price and good quality. The Cyber Commander and remote triggers will save you a lot of headaches and make flash photography simple and enjoyable. You just have to think about where to place the lights and compose your shot. A D-SLR will be best but any camera that can use a hot shoe flash should work okay. This is a much better way to get your puppy portraits than a camcorder or camera video.
If you still like to use video, then you can try the Panasonic GH2. You can adjust AF on your subject by just touching the LCD screen, to keep your subject in focus during video. It photo quality is also very good, and is quite fast (5fps at full resolution and even faster with reduced resolution). The main downside is that there is not as many lens choice for the Panasonic GH2 as the traditional D-SLRs.