Firstly, I'll explain why I don't believe in the Big Bang theory. I'll probably miss something, but never mind.
Ok, first question is when did it happen? It is speculated that Big Bang happened 13.8 billion years ago.
The age of the universe is not speculation. It is measurable.
There are two approaches. Since the universe must be older than the oldest things we can find in it, one approach is to measure the ages of the oldest things that you can find. The oldest objects with easily measurable ages in the universe are globular clusters. You can measure their ages using the fact that the stars inside them all formed at nearly the same time, and the more massive stars evolve off the main sequence first. This is the
"main sequence turnoff".
These observations indicate that the best estimate for the oldest globular clusters are 13.5 +/- 2 billion years old, with a 99% confidence level of it being greater than 10 billion years and less than 16 billion years. So this right away tells you the universe is pretty old -- at least twice the age of the Earth according to the geologists, and consistent with what you heard of it being about 13.8 billion years. But as you can see, the uncertainties in this measurement are fairly large (several billion years). We can do better.
A second method, which is much more precise, is to use the physics of the expansion. If it is expanding, then everything must have been closer together in the past. Here's the brilliant part that lets us do an age measurement
: if the expansion is homogeneous (same everywhere), and if the relationship between the recession velocity to distance is linear, then you can simply take the inverse of the expansion rate (which has units of distance per time per distance, e.g. kilometers per second per megaparsec), and you'll get the time that has elapsed since everything was together (the Big Bang).
This calculation does of course assume the expansion had proceeded from such a state, and we test that assumption (it makes a number of predictions). That is actually how we know the universe began with the Big Bang, and I'll get back to this point in a moment.
For now, let's assume (incorrectly) that the expansion rate is constant. Then the age of the universe will be the inverse of the expansion rate (Hubble's constant), which we measure to be about 68km/s/Mpc. Let's calculate that in Wolfram Alpha and convert it to years:
About 14.4 billion years. Not bad! But this is actually
wrong, because the assumption that the expansion rate is constant is wrong. It changes over time depending on the densities of matter and energy in the universe, which we can compute precisely from the equations of general relativity (specifically, the
Friedmann equations for a homogeneous universe). These equations describe the expansion rate, and its derivative (the acceleration) as a function of the size ("scale factor") of the universe, and you can use them to compute the age more precisely if you know the cosmological parameters -- which we can measure.
This is what gives us the much more precise age of 13.8 billion years.
Ok, back to the assumption of the expansion proceeding from a singularity state. This is an extrapolation of the expansion to where the universe is of zero size. That extrapolation makes predictions, because the properties and appearance of the universe as we see it today will depend on how it evolved. For example, it predicts that when the universe was younger than about 370,000 years (a tiny fraction of its current age), the density and temperature would have been so high that atoms could not have formed. Electrons would be immediately ripped from the nuclei. All the free electrons would scatter the light and the universe would be a hot opaque fog.
Once it expanded and cooled enough for the electrons to combine with nuclei to form atoms (what astronomers call "recombination" for some silly reason, even though it's the first time it happened), then the universe became transparent and the light was free to propagate through space.
This is a prediction. It has observable implications. It tells us that we should still see this light in the form of a uniform background radiation, redshifted by the expansion of the universe since that time. Indeed, this is exactly what we observe as the CMB, which was discovered quite by accident by Penzias and Wilson in the 1960s and they won the Nobel Prize for it. This was the key observation that verified the Big Bang theory over the leading alternative model (the Steady State model).
The Big Bang theory makes a number of other predictions as well. For example, it explains the relative abundances of elements, through the process of
nucleosynthesis when the universe was less than a few minutes old. It also predicts the existence of a "CNB", or cosmic
neutrino background, formed when the universe was less than 2 seconds old. But this one is much more difficult to observe since it is very weak and neutrinos are very ghostly particles.
We 'have' time, now where did it happen? It is not sure where it happened.
It happened everywhere! The Big Bang is the expansion of space itself from a small, hot, dense initial state, and it was hot and dense at all points in space. Everywhere you look, you look at something which participated in the Big Bang.
But, it 'can' be explained with Hubbles Universe and Space expantion. Where everything look like it is moving away from you.
Not just "looks like". Literally
is. This is a general relativistic effect where the space itself is expanding on large scales,
and it is readily testable. No other model has been able to explain these observations.
Now more complex question: Why? Why did it happen and also, how?
Do we need to know the why or the how in order to know that something happened? If I find a giant crater in a desert surrounded by shattered rock fragments, I can be pretty sure something violent happened there, even if I don't know exactly how or why it happened.
Sure, we would
like to know the how, but physics does not take us back before the initial singularity of the Big Bang. The Big Bang theory does not describe why or how the Big Bang originated, or what if anything happened before then. It describes how the universe has evolved from that hot dense state.
You can believe God began the universe with the Big Bang if you like -- there is no discrepancy between this idea and with science or observations.
I hope that helps explain some about cosmology and the Big Bang for you. If you still have doubts or if something is unclear feel free to ask more questions!