You can upgrade your units by unlocking their new levels at the Academy.

Tap on the Academy, and select 'Research'. The Research window will open and you select Unit. In the technology tree, locate and tap on the unit you want to upgrade, the requirements and cost will be displayed on the right panel. Tap 'Research' button on the top to start unlocking the new level of units. 

If you want to check out the stats of each unit, tap the info icon on the corner of each unit. You will be navigated to the 'Unit Stats' page.