There any number of things that could cause a flash to stop working. It could have been disabled in the menu settings of the camera, the light meter inside the camera could be faulty and not activating the flash or the gases inside the flash itself may be bad. It would be best to take it to your nearest Best Buy and let the Geek Squad take a look. They will advise you of the best action to take.

It would be best to take it to your nearest Best Buy and let the Geek Squad take a look. They will advise you of the best action to take.

Some minor repairs can be handled in the store, anything major will be shipped to an authorized repair center for that particular item.

That sounds very exciting! You should get some wonderful photo opportunities. As to what camera to get you have several choices. Canon, Nikon, Olympus and Sony all have D-SLR models under the $600 range. They all have about the same average frame rate, about 3-4 FPS (Frames Per Second). Amount of megapixels vary a lot. The Nikon being the lowest (6mp) with Sony being the highest (10.1mp).
Low light situations can be handled by changing the ISO setting on the camera. The ISO is how sensitive the sensor is to light.
I would strongly suggest that you get an additional lens. A 300mm zoom lens will allow you to get wonderful closeup shots without getting close to your subject. It will also get you a soft DOF (Depth of Field) for that blurry background look. There is some times in store specials where you get a bundle purchase. The camera with standard lens plus the 300mm zoom lens at a price under $600.
Have a safe and fun trip!