Not sure how you were able to configure more than one Layer3 interface on a 2960 switch.
A 2960 switch is strictly Layer2 and only one Layer3 interface can be enabled at the same time and this interface is used only for management, not for inter-vlan routing.
The connection from the switch to the router must be configured as a trunk interface and you need to configure all the Vlans you have above in the router. The router will perform the inter-vlan routing.
You are missing a device that is able to perform the inter-vlan routing on the Vlans you've created.
I'm not very familiar with Linux networking but in addition to the Linux kernel, the NIC must support dot1q trunking. You need the Linux box to perform inter-vlan routing for all the subnets as the switch you've purchased can't do it.
Just to verify you are indeed running a 2960 switch, can you post the output while typing the following commands in the switch:
There is another issue evident on your config which causes the inability to route to the Internet. Your "router" is on port 48 which is configured as an access port. So only devices in VLAN 1 can access it. The typical solution for inter VLAN routing is to make the connection to the router to be a trunk port so that all the VLANs can be carried over the trunk. This allows devices on all the VLANs to access the router and allows the router to provide inter VLAN routing. The 2960s are not necessarily the wrong switch - many people successfully implement inter VLAN routing on 2960s. But they need to have some device that can trunk and do layer 3 inter VLAN routing.
If they are 2960 switches then turning on IP routing is not an option. The 2960 is a layer 2 switch and does not support routing.
 though as Edison points out the configuration of VLAN interfacess 1, 5, 7, and 10 with IP addresses suggests that this might not really be a 2960. And in that case your suggestion about turning on IP routing is quite right.
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