You need VLANs if you want separate networks on the SAME router. But if you have separate routers, then you don’t need VLANs.

You will need two wireless access points. If the router you mentioned has two wireless access points built in, then just set one to connect to the shared network, and the other will act as an AP for your private network. Then the router can be configured to send WAN traffic out of the shared network AP.

If you use a router that only has a single AP built in, then you will need to purchase and additional AP to plug into one of your router’s LAN ports so that it has two total.