Welcome to the forum
Is $500 your absolute budget limit? The reason I ask, is that you could get a much, much better camera for not too much more.
The H-5 is one of the "SLR-Like" or super zoom digital cameras. They are usually pretty good cameras and can help you take really nice photos under optimum conditions. However, one big downside is that all these "SLR-like" cameras have a very small sensor. (Except the Sony R1). A small sensor means that the images will start to get noisy (grainy)...especially as you turn up the ISO setting.
A Digital SLR camera, however, has a much bigger sensor. This means better image quality and much better performance at higher ISO settings.
I don't know about the H-5 in particular, but most digi-cams I have tried have a noticeable shutter lag. You press the button and the camera waits before it can snap the photo. This is very annoying, especially when shooting sports, wildlife, golf swings etc. DSLR cameras act more like film cameras and have a very low shutter lag.
While the H-5 has a nice looking Carl Zeiss lens...that is the only lens you will ever be able to use on that camera. With a DSLR, you have the option of buying new & better lenses. Plus, lenses can be kept while you upgrade the camera in the future. On top of that, lenses hold their value very well. In 5 years, the H-5 will have been replaced or upgraded at least 5 times. It will still be a decent camera...but if you had to sell it, who would buy a 5 year old camera when new & better ones have been out for 4 years? A good lens, is still a good lens...as long as they are still making cameras to use it.
Still, a DSLR is not for everybody but at least give it some thought. For not much more money, you could really have a great camera.
Have a look at the cheapest models. The Canon Rebel XT, the Nikon D50, the Pentax K110D etc. You could even get a good deal on an older used model.
As always, it's best if you can go into a store and try out the different models. Hold them in your hands and see how they feel.
Good luck.