In connecting the switch to a router there is a question to address and that is which device do you want to do the inter vlan routing. It sounds like you have decided that you want your 2960 to do inter vlan routing. I agree with Reza that you need to check and verify that your switch does support that.
You can configure ip default-gateway on the switch. But whether it gets used or not depends on whether ip routing is enabled. If ip routing is enabled then the configured default-gateway is ignored. If ip routing is enabled then you need to configure a default route ip route 0.0.0.0 0.0.0.0 <next-hop>
If you are doing inter vlan routing on the switch and then routing to the router you will need to configure on the router static routes for the subnets used in inter vlan routing.