This should come naturally while you go for all Sub-Quests and Collectibles. There are 181 Enemies in total (110 Normal Enemies, 51 Unique Enemies, 20 Gigants).
Different areas have different types of enemies. All you have to do is enter combat with each of them. Refer to the Field Guide to see if you're missing normal/unique/Gigant enemies.
All the Unique enemies should be encountered during the main story. Gigant enemies are displayed on your map as the red diamond icon, many of these are unlocked during sub-plots.
The only areas you don’t go to during the story which contain enemies are:
Uninhabited Island
Fogwharl Limestone Caverns
The Otherworld / Earth Spirit Temple
Tower of the Wind Spirit