Question : Problem: VLAN Configuration

Our Cisco Catalyst 3560G switch has 4 VLANs created on it.  2 of those VLANs have a port connection to the 2 available interfaces on our Cisco PIX515e FW ("Inside" & "DMZ").

Does that mean members of the other VLANs on the switch can't have Internet access since none of the ports that are members of those VLANs connect to the perimeter FW?

Answer : Problem: VLAN Configuration

Configure the 3560 to route this traffic.  
Config t
ip routing
int vlan4
ip address 10.x.x.x 255.0.0.0    <-- make this the default gateway for vlan 4 devices
no shut

int vlan2
ip address 192.168.0.x 255.255.255.0
no shut

ip route 0.0.0.0 0.0.0.0 192.168.0.1

Configure PIX:
config t
route inside 10.0.0.0 255.0.0.0  192.168.0.x  <--whatever you used on the 3560

Now the pix should have a route to the vlan4

Also we need to look at your NAT statement.

It should look something like:

nat (inside) 1 0.0.0.0 0.0.0.0

If that is what it is you are in good shape if not we will have to add entries for the 10.x network.

Thanks