We weren't there; NOBODY was. There was nothing at the moment of the Big Bang, and we cannot be certain of what came into existence or when in the seconds, minutes or hours following the event.
So, "immediately"? We believe - based entirely on poorly-understood mathematical theories for which no practical test is possible - that "matter", in the sense of protons, neutrons, electrons, and their corresponding antiparticles, probably did come into existence within a few seconds of the event.
it was too hot
Neutrons combined with protons to form the Universe's deuterium and helium nuclei in a process called the Big Bang nucleosynthesis
the theory states that atoms formed during the big bang (like right as the big bang happened) Protons and neutrons were allowed to form about one second after the Big Bang. Electrons joined to nuclei to form permanent atomic bonds about 377,000 years later.
In the quark-gluon mass that was formed in the big bang light elements such as hydrogen and helium would have formed first, fusing under the high levels of gravity later.
The existence, isotropy, and spectrum of the cosmic microwave background radiation (cmbr) is extremely easy to explain if Big Bang Cosmology is true; it is impossible to reasonably explain even its existence with any alternate cosmological hypothesis.
it was too hot
Yes, it was during the Hadron Epoch.
They formed early in the big bang before matter and energy decoupled.
Neutrons combined with protons to form the Universe's deuterium and helium nuclei in a process called the Big Bang nucleosynthesis
Quarks and leptons are produced immediately after the big bang.
That depends by what level of "stick together" you are talking about. For example, quarks and gluons combined to form protons and neutrons, approximately one millionth of a second after the Big Bang. For other levels of "sticking together", check the Wikipedia article on the Big Bang.
Atoms are made up of protons and neutrons, the protons and neutrons are at the central core called the nucleus, and the electrons revolve around it, being held in orbit by the electromagnetic force between the negatively charged electrons and the positively charged protons in the central nucleus. They were formed right after the "Big Bang" when our known universe originated from pure energy some billions of years ago. From these, atoms of different elements were produced.
When Matter and anti-matter collided the Big Bang came into existence. But something still remains a mystery that how did the Matter and anti-matter come.
to find their way into the nucleas and bang with the protons.
the theory states that atoms formed during the big bang (like right as the big bang happened) Protons and neutrons were allowed to form about one second after the Big Bang. Electrons joined to nuclei to form permanent atomic bonds about 377,000 years later.
Actually, a small amount of lithium-7 also had time to form, but this fact doesn't change the basic answer. Because we know three things: 1) the probability (more precisely, the cross-section) of a specific number of protons and neutrons combining together to form an element with atomic number greater than hydrogen. 2) the density of protons and neutrons during these few seconds. 3) the length of time this "soup" of protons and neutrons was in existence. When the calculations are done, the results are that the odds of any element, with an atomic number above that of helium, forming in those few seconds is essentially zero. We can also calculate the expected density of hydrogen-1, deuterium, tritium, helium, and lithium; given the time length at which our Universe was at a specific density. The predicted density of each is consistent with the observed density. Big Bang Cosmology is unique is being able to correctly predict the ratio of primary elements in our Universe.
Some 13 or 14 billion years ago, all the existing matter in the Universe was very close together, extremely hot, and extremely dense. It started to expand, a process known as the "Big Bang", matter gradually cooled down, until it was cool enough to form neutrons, protons, later nuclei, and much, much later, galaxies, etc. It is not currently known what caused this Big Bang, what was before it (if there even WAS a "before"), what caused it, and whether this was an isolated event, or perhaps it has happened many times, leading to various universes.