To enhance a Titanio, follow these steps:

-Tap on the Shrine of Titans building
-Tap on Titanio
-Select the Titanio you want to enhance, then tap "enhancement".
-Tap the food icon to feed the Titanio and earn level EXP. 
When the Exp bar is full, the Titanio will level up
-Once you hit the threshold, you can advance the Titanio using special materials.