Ok, the resolv.conf file should contain the nameserver IP addresses of the usually two nameservers that your ISP uses. If you're using DHCP (which is most likely with a dial-up ISP) then DHCP usually configures resolv.conf automatically.
Since this is not the case, have a look in /etc/ppp or /root/.ppp after being connected. There should be a file in there called *something like* ppp0.info -- I'm not sure what the filename will be, it's been a little while since I used ppp.
If you're not sure which file, then open the /etc/ppp folder in your file browser, and WATCH CLOSELY the files in the folder, while dialing your ISP with the modem. After the connection is established, one of the files in there should either update itself, or a brand new file should appear. The filename should contain the word "info" in it.
Open that file. You should see in there, two lines (or maybe one line with comma-separated values) with the "nameserver" or "dns-server" or something like this. There should be 2 IP addresses.
Copy those addresses into your resolv.conf file, like so:
#Example resolv.conf file:
and save the file resolv.conf.
Now, try using your browser.
If this works, we'll look more closely at WHY your resolv.conf file is not being updated.