There are two ways to upgrade your cards:

After completing a Live you will see a screen like the one on the right. Your cards will get their experience from completing the Live. If a little arrow appears above your card, it leveled up. A card with a little crown on the upper right was the MVP, the Most Valuable Player, which means they got the most experience points out of all cards. In the case on the right it got 450 experience points.

That being said, experience points only raise your level (and with that your Charm and Fans), but not your Stage Appeal or Unit Skill. If you want to raise these stats, you have to use Strengthening Lessons.

Another thing that should be stated is that there is a limit on your card. This limit keeps your card from reaching its highest level (and therefore the highest possible Charm and Fans). To break the limit you have to use Strengthening Lessons again, but with exactly the same card as your Lesson Partner. In total you will need 6 copies of your card: One that you will level up and keep and 5 to break its limits.