Looking for a Good Digital Camera?

I am thinking about buying a really good digital camera that I can take really good pictures that wont blur when my daughter moves is SLR the route or is there some compact camera that is better? I am looking to spend about $600.00 to $750.00 US dollars

Blueeye...you have some good answers here. I'm going to take a different tack. The best choice for you depends on your level of interest. If you are fairly serious about photography, then a DSLR is definitely the route to go. However, they can be fairly complicated with a long learning curve...and you need to be really committed to the process, or you won't be happy with your results. Oh, and by the way, you can definitely take "blurry" pictures even with a DSLR.

On the other hand, if you want to be able to have something more compact, easier to handle and operate, take good pictures, and not have to read a 150 page owners manual before you start...look at the better compact "point-and-clicks."

There are several good models. I would recommend you get 5 megapixels or above, with a good range of optical zoom (3X or better). Sony, Canon, Nikon all have good options...it's hard to make a mistake. Expect to pay above $200-300 for a really fine compact camera.

Good Luck!

An SLR such as a Canon Rebel XT would certainly take wonderful blur free photographs if you know how to use it. It might take some investment in higher end lenses to get the results that you desire though. I would recommend looking for a SLR like camera, also called a prosumer camera that has a lens with a large F Stop(eg. f/1.4 f/1.8 f/2.0). F Stops with smaller numbers are actually larger since it is a fraction. Furthermore, the F Stop allows light to enter the lens, and will allow you to take pictures with less blur, in less light. Another option would be a better flash unit, such as an external flash on a SLR. Good luck!

Two suggestions.
Fuji Finepix F30 or Panasonic DMC FX01 camera with Advanced Mega Optical Image Stabilizer.
What you need to look for in your quest to reduce "blur" is either high sensitivity to allow fast shutter speeds which will reduce blur, or an electronic image stabilization system.
The Fuji camera has the absolute best image quality available right now in a compact digital camera under $1,000.00 I know a number of pros that use the F10, (predecessor to the F30 which is due to be introduced in June 2006) and they all swear by the outstanding image quality. The other cool thing about the F30 is it has a special flash system that provides just enough fill flash in low light situations. This gives a really nice effect and also helps reduce blur in your shots.
The Panasonic has a Leica lens that goes from 28mm to 102mm (35mm camera equivalent) so it's got a good wide angle capability. The image stabilization system is apparently quite good but the image quality is not nearly as good as the Finepix F30.
I would recommend that you check out the "bible" of digital cameras: www.dpreview.com. They have super detailed reviews of all the current digital camera models.

You can never go wrong with an SLR, but catching action shots is pretty easy for even a $400 camera. For catching action shots your camera must select a faster shutter speed. Most camera's have a little dial for selecting action shots, however that setting asks for a slower shutter speed. Why? Because with the slower shutter speed, everything NOT in motion are clear, and the object in motion has the blur to give it the appearence of movement. So just keep it in Auto.

Another thing to look at is "Shutter Lag." Maybe your camera is awsome at taking shots of objects in motion, but it takes 2 seconds to take the shot AFTER you have depressed the shutter button. Now you have to follow the object with your camera AFTER you have pressed the shutter button. That's difficult, not to mention your object might stop. So recommend a camera with short shutter lag.

Cameras to look at:
Nikon L2
Canon A540

Small, powerful, short shutter lag cameras, that are around $300.

On the other hand, the Canon Digital Rebel XT is an excellent camera. Its also a bit bulky, at a little over half a kilogram. However it has an incredibly short shutter lag, enhanced motion detection, and amazingly fast focusing (which might be critical in motion shots). But... at around $900 to $950 retail, it may be out of range.

A lot of great options listed here, but I'm going to throw another one at you.... The Sony H2. This is a new model running about $400 at Wal-Mart. It is a 6 megapixel with a 12x optical zoom. It has a 2.5" screen, image stabilizer, all the features you would expect at that price range, plus it has an equivilant ISO of 1100! That faster speed gives you an unbelievable flash range of around 45 feet. It also has the electronic view finder if she would like to shoot from a "traditional" shooting stance.