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I'm able change my network routing metrics with ifmetric, for example ifmetric enp0s3 1.

Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
0.0.0.0 192.168.1.1 0.0.0.0 UG 1 0 0 enp0s3
0.0.0.0 192.168.237.1 0.0.0.0 UG 100 0 0 enp0s8

When I reboot though, the metric for enp0s3 reverts to 101. How can I make this change permanent or have it set automatically at boot time?

The correct way to do this, in Debian and derivatives, is to write a file in /etc/NetworkManager/dispatcher.d (call it whatever you like), with the following content:

#!/bin/sh

# Change the metric of the default route only on interface enp0s3

IF=$1
STATUS=$2
MY_METRIC=1

if [ "$IF" = "enp0s3" ]
then
 case "$STATUS" in
 up) 
 ip route del default dev $IF
 ip route add default via $DHCP4_ROUTERS dev $IF metric $MY_METRIC
 ;;
 *)
 ;;
 esac
fi

This way, your customization will not be overwritten upon each update.
In order to check this, stop the Network Manager, kill the dhclient and flush the IP address of the interface, then restart network manager.

You can find documentation here.

If you are using NetworkManager, the proper way to change the metric for the default route is to modify the connection associated with interface enp0s3 in this way:

nmcli connection modify <connection-name> ipv4.route-metric 1

and then re-activate the connection:

nmcli connection up <connection-name>

You can find the value for <connection-name> in the output of nmcli connection.

You should be able to make it permanent in /etc/dhcpd.conf where you can set an interface metric like this.

interface enp0s3;
metric 1;

The easiest and right way to do this is by editing /etc/network/interfaces.

Here is a simple example of /etc/network/interfaces:

auto lo eth0
iface lo inet loopback

allow-hotplug eth0
iface eth0 inet dhcp
 metric 700

Restart networking using service networking restart for the changes to take place.

Ref: Section 5.7.2. The ifmetric package of Debian Manual

The following sets the eth0 interface to be preferred over the wlan0
interface.

Install the ifmetric package.

Add an option line with "metric 0" just below the "iface eth0 inet
dhcp" line in "/etc/network/interfaces".

Add an option line with "metric 1" just below the "iface wlan0 inet
dhcp" line in "/etc/network/interfaces".

Extra Note:

• 
  auto <interface1> <interface2> starts interfaces on start of the system.


• 
  allow-hotplug <interface> starts the interface on hotplug event.