I consider myself a digital camera enthusiast, and I'll give you two pieces of advice.
First, If staying within your budget is more important than getting clear pictures of your dog, then in my opinion, the best deal by far in digital cameras is the Canon A570IS:
Amazon.com: Canon PowerShot A570IS 7.1MP Digital Camera with 4x Optical Image Stabilized Zoom: Camera & Photo
It sells for about $150 online, and it takes really great pictures. The camera has a large LCD screen and an optically-stabilized lens that will help you take low-light pictures without a flash (although it won't freeze a moving dog). This camera will take great pictures, especially in good lighting conditions and where the dog is not moving too much. It is a point-and-shoot camera, meaning it has a flash and lens built in, and is pretty small.
My second piece of advice: if getting better pictures of your dog is more important than staying within your budget, you really really really should consider getting a digital SLR. A digital SLR has the following benefits over a point-and-shoot camera:
1. The imaging sensor is bigger, which gives you low-light pictures with less grain and noise.
2. The focusing is faster for better action shots, and you can hold the shutter to take a burst of shots to make sure you capture the right moment.
3. As your skills develop, you can get different lenses for different situations that will make your pictures really stand out.
4. You will be able to avoid using a flash, and therefore avoid getting those dreadful "puppy eye" photos that make your dog look possessed.
A digital SLR like the Canon Digital Rebel XT will cost you about $450 with a lens:
Amazon.com: Canon Digital Rebel XT 8MP Digital SLR Camera with EF-S 18-55mm f3.5-5.6 Lens (Black): Camera & Photo
I know that this over twice your budget. But take it from someone who has bought three point-and-shoot cameras trying to find one that could capture dogs best: a digital SLR is the only camera truly well-equiped to capture all the moments you'll want to remember.
The Rebel XT is a really good deal in digital SLRs. The Nikon D40 is another really good deal, and you can get the camera plus two lenses for about $450 (I just happen to prefer Canons, so I included a link to the Rebel XT).
Here are a couple examples of pictures I took of my dog, Buster, last night using a digital SLR, which, because of the low light, I KNOW I could not have caught with a point-and-shoot: