Where were the first galaxies formed in our observable Universe, at the center or at its outskirts? After inflation where were the first galaxies formed? At the center (our home position) of our observable Universe or at its outside rim?
I know that for the case of a galaxy the standard view is that the first stars were formed at the outskirts of a galaxy. Is that the case also for our observable Universe concerning the galaxies formation?
If the first galaxies were formed at the outskirts is that because this layer cooled off first?
 A: Our universe does not have a “center.” The distinguishing feature of our location is that we live here.  We have excellent evidence that the universe is “isotropic,” which means the time from the Big Bang to galaxy formation would have been the same everywhere.
For an analogy: where is the center of Earth’s surface? Is it one of the poles? Latitude zero, longitude zero? Your house? My house?  The Earth has a center, but its surface does not.
A: The following  is a good analogy for the current big bang cosmological model as far as the answer to "where the center of the universe is".
Take an ideal balloon,before inflating it, its center  at (0,0,0) and all its pre-inflation mass sitting there(thought experiment) . Inflate it at (0,0,0) to a spherical surface , an ideal sphere at a radius r. All points on the surface were at (0,0,0) at time 0 before inflating the balloon. There is no center on the surface of the balloon, all points are equivalent.
It is the two dimensional analogue of the Big Bang model, the current model of cosmology, which is in three space dimensions and one time dimension, expressed in four vectors.  The hypothesis that the universe started expanding 13.8 billion years ago from a singularity fits well enough all the observations at present and so , all present points of the universe were at that original expansion point and can be considered the center of the observable universe.
Now in the comments you ask:

I think my question is pretty clear and straight forward. You have a volume of hot vapor and a temperature gradient. Were will the vapor cloud first start to condense? Near its outskirts or near the center?

In the balloon analogy , there will be different densities on the ball0on surface, originating at different times dependent on the thermodynamics and elasticity  of the surface. Analogously in the three dimensional space of our universe there will be different densities originating at different times, but not dependent on the location, just on the dynamics of the system, densities etc.
A: If I understand your question correctly (based also on your disagreement with the interpretation of the other answers), what you are asking is whether the first galaxies were formed at the particle horizon "outskirts" that delimit the observable Universe at some earlier point or somewhere within the current confines of it. If that is indeed the question, then the answer is the former.
Quantum fluctuations of the scalar field whose potential dictates inflationary expansion (usually called the inflaton $\phi$) during the slow roll regime correspond to physical scales which we may denote as $\lambda$. You may think of $\lambda$ as the "distance" a quantum fluctuation has and thus where it's located in the initial inflationary patch. These scales are subject to inflationary expansion as well, since the quantity $a\lambda$ is the physical wavelength of the perturbation in the comoving frame of reference.
If a certain physical scale $\lambda _{0}$ appears at $N_{0}$ e-folds after the slow roll started, then the physical wavelength by the end will have grown:
\begin{equation}
a(N_{\text{tot}}) \, \lambda _{0} \sim a(N_{0}) \, \lambda _{0} \, e^{N_{tot} - N_{0}}
\end{equation}
where $N_{\text{tot}}$ the total amount of e-folds of the slow roll. If the e-fold difference and initial scale of the fluctuation have an appropriate size so that $a(N_{\text{tot}}) \, \lambda _{0} \gtrsim H^{-1}$ ($H^{-1}$ being the initial - Planckian - size of the particle horizon), then said physical scale crosses the horizon and remains "frozen" beyond it as soon as the crossing occurs. "Frozen" here implies the corresponding (quantum fluctuation) inflaton dynamics become static.
After inflation and the subsequent reheating phase conclude, the observable Universe will keep expanding in a radiation-dominated manner, so the particle horizon $H^{-1}$ which was roughly constant during inflation will begin to increase. As it increased and approached its current size (around $3000 \, Mpc$), physical scales that crossed the horizon during inflation close enough to the end of inflation will be able to cross back in the horizon. Since they are now macroscopic in size, they are imprinted on the spacetime as density perturbations that cause local inhomogeneities. This is precisely the nature of structure formation by which galaxies are formed.
Since in this paradigm of structure formation it is inflation-era quantum fluctuations that are responsible for galaxy formation by crossing out and back in the particle horizon, galaxies are inevitably first formed close to the limits of the observable Universe at each point where a particular crossing occurs.
As an addendum, to make a connection with anna's balloon analogy, you can picture galaxy formation like this: Imagine you have a balloon that begins very small and we're at its surface. During inflation, this balloon remains roughly of equal size, but an "imaginary" balloon space where the current balloon can expand into grows immensely. Our balloon experiences small perturbations on the surface that also grow immensely, so they stop acting as perturbations as soon as they become bigger than the balloon itself; they enter the imaginary balloon space.
At some point this phase ends and our balloon starts inflating at some (smaller) rate. As it inflates and encapsulates more and more of the imaginary balloon space that grew earlier, it "bumps" into those earlier grown perturbations. Upon bumping into them, the density of the balloon changes and structures (your "balloon galaxies") form on the surface.
