Your networks need to be "reversed."
Presently your new network with your Raspberry Pi is connected to your old network with your other devices, which in turn connects to the Internet.
Instead, you should have your Raspberry Pi (which you want accessible from the Internet) in the network that's the closest to the Internet, and the rest of the devices in the second network. In other words:
Internet---Router 1---(Raspberry Pi) | +---Router 2---(other devices)
Set up this way:
- All of your nodes will have access to the Internet
- A port-forwarding rule on Router 1 will make the Raspberry Pi accessible from the Internet
- The first network (with the Pi) won't have access to the second network due to Router 2 firewalling that network