Whoever recommended for you not to put your LAN on pub IP's was right, as are the other posters.
Multiple private IP LAN clients can "share" a single public IP for internet access via NAT, i.e. Network Address Translation. Most routers have this capability.
A simplistic description: the router takes the outbound traffic from your LAN and replaces the source internal private IP's with the router's public IP. The router keeps track of which internal IP's are talking to which external IP's so that the inbound traffic gets sent to the correct internal IP. This allows multiple internal clients on private IP's to communicate accross the internet using as little as a single public IP on the router's external interface.
This by itself offers a certain amount of protection as your LAN clients are not exposed on public IP's and inbound traffic to the NAT external IP is generally only allowed based on a request from an internal client. For more protection, also install a firewall.
So to make this work on your network, you would
a) configure your LAN clients back on Private IP's, properly configured with your router as their default gateway, and with working DNS resolvers;
b) configure your router's internal interface on the same network as your LAN of course, at least one Public IP on its external interface, and enable NAT