Some times I feel like if my eyes were a camera, I could capture exactly what I feel in my eyes, the awesomeness of the night view in streets under street light. But I can't. Because I am not a expert in photography, whenever I take night shots, it miss everything. If someone could help me how to take night street shots under street lights, it would be so helpful. I have an Olympus FE 210 digital camera. And also could someone suggest a cost-effective camera for someone who is not a beginner, but not yet an expert?
Actually, Even if there were a camera with the capabilities of your eyes, the photos you could take with it would be really shitty. What makes your eyes awesome is the software, i.e. your brain that does a ridiculous amount of incredibly sophisticated post-processing that uses time sequences of sensory input to simulate a much higher resolution and dynamic range than the hardware actually has. Heck, it even fills in stuff that you don't actually see! Unfortunately, it doesn't work on still photographs.
And don't take this as criticism, but your question clearly shows that you are an absolute beginner in photography. To change that, you need to learn about the exposure triangle.
Essentially, the problem is that photography needs light, and in night shots there is very little light available. And compact cameras like yours are bad at dealing with that situation. With some compact cameras, you can work around their weaknesses by setting a really long exposure (5 seconds or more) and using a tripod (or makeshift support) and the autotimer for release.
But that only works when the subject is static, and the Olympus FE 210 doesn't even allow choosing a longer exposure, which means it's basically useless for night shots.
So what you really need is a better camera, i.e. one that allows control over exposure settings, has larger aperture and higher ISO with less noise. And a tripod.
My suggestion would be a low-end mirrorless system camera, and getting a fast prime lens for it. I'm very happy with the NEX-3N and the Sony F1.8 35mm lens. This is my favourite low-light picture I took with it (shot handheld, with no illumination except those sparklers):
That lens is a bit pricey, but there are cheaper alternatives that are almost as good, and for now the kit lens that comes with the camera may already be enough; certainly much better than your compact camera.