Many/most routers allow will forward DNS requests to the ISP's DHCP-assigned DNS servers (because its WAN config will know what DNS servers your ISP's DHCP server told it to use). That means that you could just use the same address for Primary DNS Server as you do for Default Gateway (10.1.1.1).
Alternately, you can use public internet DNS servers, if you don't want to go to the trouble of finding your ISP's. 128.101.101.101 and 134.84.84.84 are common ones that I know people around the internet use, because they're easy to remember (they're run by the University of Minnesota and they've been around for ever, as a side note).
I'm pretty sure you don't want to do what vee_ess says. Static routing is something different from port forwarding and that stuff. If you don't know what that stuff means, I'm not going to explain more than to say that you don't want to mess with it.
To clear up a few things (hopefully), about DHCP vs static IP assignment:
If you're only running a couple computers on your LAN, you're probably just fine to leave the router's DHCP server enabled, but statically assign your computer's IP address. If you want to be even safer, assign it an IP address that's outside the scope/range that your DHCP server is set to assign (it might be set to assign 10.1.1.2 - 10.1.1.25 or something, so you could use 10.1.1.26). Many DHCP servers will cache the entries that it has assigned in the past, so when your computer gets assigned an address once, it will get assigned that address every time it requests an address. This is not inherent to how the DHCP system works, though, and you /shouldn't/ depend on this, ideally.
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