Within the Research menu, you will see an Upgrade button in the middle of the screen. If an upgrade is not available, the button will be greyed out.
On the Upgrade screen, select the card you wish to update. Then select the card(s) you would like to use in the upgrade. These cards will be consumed towards the upgrade of the initial card you selected. If the card reaches the next level in the upgrade, then you will see that its attack and defense stats have been increased.
Starter cards cannot be used towards the upgrade of another card until you reach level 30. This is to ensure your deck has enough starting combos to play in battle during the early stages of the game.