First of all, lets see if we can straight things out. A device that has an IP address is either a host ( your system ) or a router, the thing you are calling Modem. Modems modulate and demodulate, they don't route and have IP addresses.
I would bet you have a dual purpose machine, modem/router. This is common these days. The modem part just adapts to the media you have, ADSL or Cable.
You should be able to point your web browser at it, type the IP addess in the location bar, press enter, and get to it. Try the gateway address. Most are password protected, and all too many people never change the default factory password. Great for people wanting to use your service for free, or hack your network.
The ISP side of your router will have a IP address assigned by your ISP, most, but not all ISP's use DHCP to assign an address. The rest require you to configure in the ISP side IP address.
An address on 192.168.xxx.xxx is a private address, and gets translated in your router. Don't worry about that, it will work unless the router is broken.
It sounds like you assigned the IP address to your linux system. Is that correct? If yes, you need to look at the router config to see what range of addresses it is configured for, and make sure you are part of the same sub-net. If you are not, then the router will not route your packetts. The masks look good, that is a standard class C mask.
The primary DNS IP address doesn't look right to me. Most routers will aquire the DNS addresses from the ISP, and pass them to the systems using the router using DHCP. If you assigned the IP, you need to find out the correct DNS address from the ISP, or your router. Mine displays the DNS addresses. Once again, look at your router config to wee what you can find out your self. Ask your ISP if you can not see the info in your router.
Valid addresses for the internet will not be private IP addresses such as 192.168.xxx.xxx. Your side is a private network and an address of 192.168.xxx.xxx there is O.K.
You will have a resolv.conf file some where on your system, usually in /etc directory, or it may be a sim link to the real resolv.conf file. Have a look at it and see what addresses are in it. As an example only, here is mine.
Code:
cat resolv.conf
nameserver 67.69.184.148
nameserver 206.47.244.56
/sbin/ifconfig in mine shows:
Code:
/sbin/ifconfig
ath0 Link encap:Ethernet HWaddr 00:11:95:4B:F8:28
inet addr:192.168.1.14 Bcast:192.168.1.255 Mask:255.255.255.0
inet6 addr: fe80::211:95ff:fe4b:f828/64 Scope:Link
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:76660 errors:35745 dropped:0 overruns:0 frame:35704
TX packets:67010 errors:0 dropped:1 overruns:0 carrier:0
collisions:0 txqueuelen:199
RX bytes:22249799 (21.2 MiB) TX bytes:8770567 (8.3 MiB)
Interrupt:10 Memory:cc540000-cc550000
Note the IP address, I'm running DHCP, my router assigns the address. The third field is the sub-net. In my case it is 1, yours is 0. That is OK as long as it matches your router config.
Have a look at things, and post pack what you can discover. It helps if you post as much info as you can gather, such as your /etc/resolv.conf file, /etc/hosts file and the output of /sbin/ifonfig.
Hope this helps.