Welcome!
If I understand correctly, you want to plug the AirPort Extreme Base Station into the Linksys gateway via ethernet so that you can use the AirPort's wireless functions (as well as its wired ports). There shouldn't be any problem with this, but there are a couple of tricks. First, plug into one of the AirPort's LAN ports and using the Network preferences pane, choose the Built In Ethernet adapter, select "TCP/IP" and "renew the DHCP lease". This will put your adapter in the same subnet as the AirPort device and let you communicate with it. Now you should be able to use the AirPort Admin Utility to configure the Base Station. You'll want to tell the Base Station to receive its WAN IP automatically.
Now, plug into the WAG and repeat the "renew lease" procedure. You should now be able to get to its configuration page and set it up as well.
Now the trick: if you want everything on both the Linksys' and AirPort's networks to be able to communicate together you need to make an adjustment in how one of 'em uses addresses because they default to different (VERY different) sets of addresses. My advice is to let the WAG remain "in charge" of all the addresses. This requires you to simply turn off the "Network Address Translation" function of the AirPort Base-which is as simple as unchecking the "share a single IP" box in the Network tab of the Admin Utility.
Summary: Configure the AirPort Base first, so that it will automatically get its WAN address from the Linksys gateway (later you may want to set the Base so that it has a fixed address, but that's later). Next, plug into the gateway and configure it. Finally, configure the AirPort Base to pass on everything by unchecking the "share a single IP" box. This will allow you to put any and all of your computers on the same network so that they can see each other and share files, as well as getting online to surf.
Let me know if this was helpful, and again WELCOME!