Problem is, the 2960 series of switches from Cisco are layer 2 only switches, so if you want to create VLANs, that's fine, but you must have a device that can route those VLANs your create. This can be done with a traditional router, but my suggestion is get a Cisco 3750 series switch to use as your core switch. This is a layer 3 switch so you can create your different VLANs on this switch, enable ip routing on the switch, propagate those VLANs down to your 2960 switches via VTP, plug your servers into the core and nodes into the 2960's which become your access switches. The 3750 switch will be your default gateway/router for your network, then you can establish a route from it to your firewall/router to Internet.
I am not a Cisco product guru, but I believe these switches are also capable of being stacked together, so if you need to increase redundancy, you can get two, stack them together, run two uplink cables from each of your 2960's to each of the 3750's in the stack. Most servers come with dual if not quad NICs, so the same can be done there also. This way, if you have a switch go down, your still up as the other switch is running and you have redundant links from your servers and from your 2960's.
Hope that helps.