Attacking Beasts

Attacking Beasts is not an easy task, indeed it can become one of the tougher challenges you will face! 

Pay attention to Beast's levels. On the map, you can find Beasts with levels 1 to 20. A Beast can only be attacked once you have already killed at least one Beast of the level below it. Of course, Beasts of higher levels will offer better kill rewards.

Finding targets
The easiest way is to use the Hunt button - - when browsing through the map. It will bring up a screen with Beasts that are on your highest Beast kill level that you have gotten to, as well as 1 level below, and 1 level above:

Recommended March Power
The Power of your attacking march has to be higher than the power of the Beast you attack, if you wish to win. The march setup screen will warn you if your power is not enough. If your attack anyway, you risk high casualties.
It is recommended to attack with a Guardian that by itself is powerful enough to overcome the Beast. In doing so, you can send only a single troop with the Guardian, limiting your casualties to zero.

Killing a Beast
If you manage to kill a Beast, you will receive some loot. If you are lucky, the dying Beast will also spawn a Shrine.

Shrines are unique locations where you can collect Elixir. Shrines appear after killing a Beast on the map (not always, but there is a chance):

The higher the level of the Beast the more Elixir the Shrine will contain. To collect the precious resource, you will be required to send some troops and the Guardian of the same "species" as the Beast you just killed.

In the situation of not mastering the required guardian, you should tell one of your allies to benefit from the Shrine. All the alliance members will be awarded a gift every time a member empties a Shrine, so don't let them go to waste!

You can technically farm shrines that were spawned by other players' kills, but keep in mind that some players may feel territorial about them, so you may face some retaliation.