What is basic methods of use Barbarians and Archers?

Upgrade your archers as much as you can. Push them and let your barbarians clean up the rest. If needed use giants/balloons (depending on your Barack level) to take out the mortar and archer towers so it is easier for the archers to push. Be careful as the mortar can easily take out archers if you don’t place your Troops right. Take note, as you level up your town hall and barracks, you should only be using archers, not barbarians.

The basic method with the use of barbarians and archers in attacking is to let the barbarians go first to attack the defensive buildings or walls followed by the archers behind the barbarians.
The barbarians should go first because they have larger hitpoints and can endure longer in battles compared to that of the archers.
These barbarians will act as shield to the archers that are behind them targeting the enemy’s defense buildings, collectors, or storages.
This type of attack is used in attacking enemy bases for loots especially if the storages are empty and most of the loots are concentrated in the collectors.