bridge vs iptables vs route?

Bridging is for adding/combining segments to a network. When you bridge two network interfaces, what you are doing is allowing network broadcasts to reach all of the nodes. Bridging would not apply to what you are trying to do.

For the router to function, here is a basic reference:

check ip forwarding status for routing:
sysctl net.ipv4.ip_forward

set (enable) ip forwarding status for routing:
echo 1 > /proc/sys/net/ipv4/ip_forward

unset (disable) ip forwarding status for routing:
echo 0 > /proc/sys/net/ipv4/ip_forward

After you have verified that ip forwarding is enabled, you can use iptables on the public interface, the network interface with the physical internet connection, as a NAT interface.

I won't attempt a comprehensive explanation of NAT here, but this is a basic example:
echo 1 > /proc/sys/net/ipv4/ip_forward
/sbin/iptables -t nat -A POSTROUTING -o eth1 -j MASQUERADE
/sbin/iptables -A FORWARD -i eth1 -o eth2 -m state --state RELATED,ESTABLISHED -j ACCEPT
/sbin/iptables -A FORWARD -i eth2 -o eth1 -j ACCEPT

you can see the nat like so:
sudo iptables -L -t nat