I am able to ping my a computer on my network from my Galileo. I did nothing special, I just let the DHCP server assign the Galileo an IP address (and gateway, and DNS, etc).
It looks to me like your routing is not correct? To me (not a Linux expert) like you are missing an entry for your Gateway?
Secondly, if you are trying to install packages on Galileo you might have trouble. There is no apt-get on the standard image. I think you can build a full image that will include an installer.
The ping works in both directions on Debian, but I still cannot access the Internet from the Galileo. I tried to make my PC as a router with the command:
and I also added a default route on the Galileo to my PC (just like Jacker.Wang says), but it doesn't change anything. Is there a better way to access Internet from the Galileo through the PC?