The “Big Bang” is a model, used to explain what we can observe in the Universe. It makes predictions and it can be used to explain a lot.
But it is a MODEL. It is not a fundamental truth. (See my answer to this other question http://ias.im/42.1143 ). As a result, you need to be careful using words like “believe in” (which makes it sound a bit religious).
It is also a model with some problems. Issues of the singularity and the implications of infinite energy densities, problems to do with asymmetry and other issues mean that there are still some things that don’t seem quite right with it. As Tim points out, there are groups trying to work out what the limitations are, and improve the theory.
And also there are other cosmological problems (such as dark matter and dark energy), which also don’t seem quite right. Maybe the nature of the Big Bang needs more work, to help resolve these problems.
However, despite these criticism, the Big Bang Theory does a good job of explaining much of what we can see and measure. I think it does a good job of explaining the bulk expansion of the Universe and the age of the stars and galaxies in it.
I certainly use it for my everyday work and rely on it for calculations.