Why is a lower Noise Figure considered better than a higher Noise Figure?
o does this mean our final objective is to always try to make our SNR(out) as high as we can 


Yes, we want the output SNR as high as possible, because that means the best chance of recovering the message signal accurately.


  so that our signal is amplified much more than our noise?


This is not possible. 

The input noise will be amplified just as much as the signal.

Plus, some additional noise will be added by our amplifier.

So the overall effect is the output SNR is lower than the input SNR. (This is why the NF is positive when expressed in dB)

But we want the SNR to be reduced by as little as possible.