Basic of a charged piezo : pC/G, mV/G, Signal analysis
How to read a piezo signal like the signal on the scope? Should I read only the first Edge from the signal like in the image below?


That depends on your application.  I read it as an amplifier that's saturating for the first \$80\mu\mathrm{s}\$ or so, because the gate voltage is wiggly but the FET output is sitting on ground.


  What is pC/G ?


Pico Coulombs per g, with \$\mathrm{1g = 9.81 m/s^2}\$.  \$0.84 \mathrm{pC/g}\$ means that for a sudden 1g change in acceleration, 0.84 picocoulombs come out (or go into) the piezo device.  Typically you'll have the thing loaded with a bit of resistance that'll return its voltage back to zero (or whatever you've biased it at).


  How to make a conversion pC/G to V ?


If the piezo has a well-defined capacitance (look in the datasheet) then within its bandwidth the piezo itself will output a voltage of \$k / C_p\$, where \$k = 0.84\mathrm{pC/g}\$ and \$C_p\$ is the piezo's capacitance.  In that case, you could follow it with a plain old voltage amplifier of the appropriate gain and bandwidth.

The datasheet I found lists an output capacitance of \$390\mathrm{pF} \pm 30\%\$.  That's a pretty large range.  Add that to the sensitivity variation of \$\pm 15\%\$ and if you depend on the output capacitance for accuracy then your overall variation is somewhere around \$\pm 50\%\$.  That means you either make inaccurate measurements or you calibrate each sensor.

If you want to increase accuracy, then you'd need to make an amplifier with a low input impedance through the range of frequencies that you're interested in, and make sure it's stable.  There's not nearly enough information in your question for me to know if that's necessary or desirable.


  Is my Amplifier even correct? Any other low-cost suggestion ?


That depends on what you want out of it.  I'm not sure that, with a 5V supply, you can even count on that working for every transistor you put in there, or over any sort of temperature range.  I'd be far more inclined to use an inexpensive op-amp in non-inverting mode, but your "low cost" may be lower than mine.