When the star formation begins? We can separate the history of the universe in different epochs. 
Radiation dominated epoch, matter dominated epoch, and dark energy dominated epoch, and we can divide the epochs in different ways.
For example the radiation dominated epoch can be divided in the epoch before atoms could form and the epoch after the first atoms form (nucleosyntesis) my question is, when the first star form?
What is the aproximated redshift where the stars star to formed?
(I think that is a really high redshift because the observations of the star HD 140283)
 A: The first stars formed 180 million years after the Big Bang
Although hydrogen becomes neutral at recombination, 380,000 years after the Big Bang, density fluctuations at this time are much, much too small to form any structure; typical fluctuations are $\sim10^{-5}$ denser than the average.
It is true that the overdensities at this time are the seeds for later structure, as they are gravitationally amplified with time. But we will have to wait another 180–200 million years — corresponding to a redshift of $z\simeq20$ — before the first stars form.
The first stars were very massive and very luminous in the ultraviolet — their light was able to significantly alter the state of the neutral hydrogen's hyperfine line at 21 cm$^\dagger$. This made the hydrogen able to absorb a fraction of the cosmic microwave background, and this absorption was observed recently by Bowman et al. (2018).
The star HD 140283 is not older than the Universe
The age of HD 140283 was determined by Bond et al (2013) to be "$14.46\pm0.8\,\mathrm{Gyr}$". This is sometimes (in popular literature) thought to pose a problem for cosmology.
But, firstly the measurement involves uncertainties in parallax, stellar parameters, chemical composition (especially oxygen), and probably more I don't know about.
Secondly, if HD 140283 formed 180 Myr after the Big Bang, its age would be $\simeq 13.619\,\mathrm{Gyr}$. This is only $1.05\sigma$ away from the estimated mean, so I would say it's quite consistent.
The authors don't claim the star to be older than 13.8 Gyr, they just quote this (quite uncertain) result, and say that the only thing it implies is that it must have formed soon after the Big Bang.
Uncertainties are not hard limits, they (typically) refer to the standard deviation, meaning that $14.46\pm0.8$ implies a 68% probability that the true value is within $14.46\pm0.8$, a 95% prob. that its within $14.46\pm1.6$, 99% prob. it's within $14.46\pm2.4$, etc.

$^\dagger$More precisely, the hard UV was able to ionize the hydrogen in the vicinity of the stars. The hydrogen quickly recombined, producing copious amounts of Lyman $\alpha$ light which, in turn, were able to make the spin temperature follow the gas temperature, which at this time is significantly lower than the CMB temperature, producing a strong absorption signal. See e.g. the review by Pritchard & Loeb (2012).
A: This is the history according to the mainstream model, the Big Bang

After the decoupling of photons at 380.000 years, there is  a period marked as "neutral hydrogen forms"  that carefully is not given a time to the line of the next stage, which is modern universe. In other plots of timeline for the Big Bang the line is drawn at $10^{{17}}$second
Once matter is mainly neutral, gravity is the main force remaining , and is stronger than the gradual, even if accelerated, expansion. Masses will gravitate statistically , (the hypothesis is), toward the denser regions seen in the cosmic microwave background CMB , thus starting to form the kernels of stars galaxies and clusters of galaxies.
As it is a statistical phenomenon, dependent on the density, which was  created with quantum fluctuations during the inflation period, one cannot think that there exists a specific line that separates star formation from hydrogen etc soup in the last interval. There is a probability that stars formed in certain regions very soon after hydrogen became neutral.
Of course all this within the model,which is at present mainstream.
edit after discussion:
The process from the time of neutral hydrogen  to the present observable universe seems to be under research:

"Exactly how stars and galaxies formed, when the process started and how long it took is currently a major area of research. A simple picture runs like this: about 1 billion years after the big bang the first star forming regions, conglomerates of perhaps 106 to 109 solar masses began to develop. Over the next several billion years, most of these merge to form larger units or are partially destroyed by the energetic supernovae which develop as a natural part of star formation. Within a few billion years most of these have developed into stable configurations of stars and gas and are recognisable as ``galaxies''. 

