What is the best way to feed ego taming?

In order to feed your monster in Ego Taming, you need to provide it with the correct items. You can provide different weapons, Armor, Substitute, and accessories of rare and epic rarity only. This will be needed later on as you progress, since the stats that this provides will aid on your strength and survival. You can find these items from the following:

Crafting Tier III items above

Once you have unlocked Tier III recipes and above, you should be able to create items of the rarity that's needed. Having extra materials from the Material dungeon will let you create more items that you can feed.

Item Rewards from Chapter 2-3 onward in Adventure

Starting from Chapter 2-3 onward, rare items will be available as a drop by chance. So depending on your progress, farm repeat chapters of the latest one that you can, this is a good way of getting gold along the way as well.

Rings and Amulets from the Honor Shop

The Honor shop lets you use your honor coins and get rare/epic rings and rare/epic amulets. Use this if you already have enough equipped on your characters, and if you can afford to buy extras.