Charasteristic curves for a transistor working in a common emitter configuration
I can't understand the reason for Why the R out should greater than the R in and what is the expectation of that?


Roughly, the voltage gain of the CE amplifier is

$$A_v = \beta\frac{R_c}{R_b}$$

So if \$R_b\$ is too large or \$R_c\$ is too small, the gain is low.

I'll answer the 2nd and 3rd questions out of order:


  I can't understand the reason of why there is a saturated region after the activated region. Why does the activated region can't to be continue.


If the curves for the active region continued to the left of the y-axis, that would be in the 2nd quadrant of the graph. This would require the transistor to provide power to the rest of the circuit. But a transistor is not a power source, so it can't do that.

I've answered this in more detail in response to past questions: 1 2.


  What is the reason for have two gradients? Starting have a speed variation and then slow gradient... I tried to concentrate it with the Ohm's law but couldn't.(According to the ohm's law the R out should different at the two situations.)


Two modes of operation, two different behaviors. 

If it's clear the transistor can't continue the active-region behavior into the 2nd graph quadrant, then something has to happen to keep the curve in the 1st quadrant. What happens is what you see in the graph you included in your question.