After you get the Rumble Badge from Korrina, she asks you to battle her at the top of the Tower of Mastery.
Climb the Tower
Go back into the Tower of Mastery. The guru will say that Korrina (a.k.a. the Mega Evolution successor) is at the top of the tower. Go north to the back of the room. There is a Black Belt standing there. He will see that you're approved to climb the tower. Go east up the ramp now.
You'll pass by a big door. The door just leads to a room with a couple of people who will say some things about Mega Evolution. One of them tells you the guru's real name!
Continue going up the ramp. You'll pass by another door, where there are a couple of people and a Lucario. You can learn more about the guru and Mega Evolution from them.
Keep going up the ramp. In the next room, the purple-haired guy gives you TM47 Low Sweep.
Go up the ramp again and you will reach the last door. You'll find Korrina with her two Lucario. Korrina will give you the Mega Ring. It's the size of a bracelet, so you put it on your left wrist. Korrina says to give your pokémon the correct Mega Stone, and you can use the Mega Ring in battle to Mega Evolve that pokémon.
The Lucario that likes you will come up to you and want to battle alongside you. Although Korrina is shocked that her Lucario has a stronger bond with you than with her, she accepts it and asks if you and Lucario will battle against her and her Lucario. They both have Lucarionite, which is the Mega Evolution stone for Lucario. If you're ready for the battle, say yes.
Mega Evolution Battle
Successor Korrina has a level 32 Lucario. Your Lucario is also level 32. To Mega Evolve your Lucario, first tap Fight, then tap Mega Evolution at the bottom of the screen. If the Mega Evolution button is lit up and glowing, it means that your pokémon will Mega Evolve after you choose which move it will make.
To win this battle, first tap the Mega Evolution button to make it glow. Then use Power-Up Punch, and use it again on the next turn (don't try to heal; just attack).
Your Lucario is faster, so it will go first on each turn. You should be able to defeat Korrina's Lucario with only two Power-Up Punches. Don't try healing up your Lucario. This will put you at a disadvantage and you might have to heal up over and over until you get a chance to attack.
Korrina will congratulate you and will say that you should take Lucario with you. If your party is full, you'll be asked to choose a pokémon in your party to send to the PC Box. You'll be asked if you want to give your Lucario a nickname.
After that, you should leave the tower.
Go east from the Shalour City Pokémon Center and talk to the hiker near the sign. In exchange, he'll give you a Sun Stone.
Continue going east toward the exit that goes to Route 12. Your neighbor will run up and give you HM03 Surf. Continue going east to reach Route 12: Fourrage Road.