Schools subnet = 10.0.0.1 Classes subnet 10.0.0.1 can't connect to internet

Some details are missing, so I'll do some guessing here:

Is your "dlink network" set up with the D-link as a router, so that it is the only gateway to the rest of the school network? (Should be, to isolate the networks.) If it is, does it do NAT? If not, you have to make sure you don't distribute the same IP ranges from both units. (The success when you use a different private range suggests that this may be part of your problem.)

If your Dlink is set up as a NAT router, it may have a static external address, probably in the 192.168.x.x range. For it to work you need to set the external address to somewhere in the "linksys" ip subnet and outside the linksys DHCP range, or you could set it to obtain IP automatically (from DHCP), if that's an option.

Think of the subnets as streets. You have two "streets" with the same name. How do you get from 123 Elm Street to 145 Elm Street? Are they really on the same street, or on two different streets that have the same name? Honest and true - computers can't read your mind - you have to tell them what to do. When the "street names" (networks) are not the same, there is no confusion of where to go. However, for this to work, hosts on both networks have to know how to reach the "other" network. I suspect your router is NATing the addresses on "your" net, so that they appear to be coming from a single host on the "other" net.

Well, that might be, but does the host on the internet know how to send packets back to you - so that you can have a conversation, or is that host just receiving packets from some unknown computer that it doesn't know how to talk to. At every step from "here" to "there" everyone has to know how to send packets to "here" and to "there" If it can't find "here" or "there" there is no conversation - no connection.

Old guy