Ubuntu, two NICs but only one with internet connection

Solution 1:

As pointed out in the above comments you have two default routes (via 192.168.1.1 and 10.0.0.1) but only the first one connects to the Internet. If the routing table remains as you posted you will have an Internet connection: whenever you have two routes for the same destination, the one with smaller metric wins.

However, since the metric is not assigned by you, but by the DHCP clients, the value of metric depends on the order in which the interfaces are brought up. I would change your configuration to a static one. Add to the file /etc/network/interfaces the following lines:

auto enp2s0
iface enp2s0 inet static
    address 192.168.1.2
    netmask 24
    gateway 192.168.1.1

auto enp2s1
iface enp2s1 inet static
    address 10.0.0.5
    netmask 24

and restart networking (sudo systemctl restart networking).

PS: This is the oldest way to configure networking on Ubuntu/Debian. On a new Ubuntu there is certainly also NetworkManager and systemd-networkd (in order of appearance). There is also an equivalent configuration for those two.