From Sonic Retro
There are three different colours of Power Core: blue, yellow, and red. The Blue Power Core levels up the Speed character, the Yellow Power Core levels up the Fly character, and the Red Power Core levels up the Power character. It take three Power Cores to level up a character fully.
Power Cores can be found in various ways. One is to find them in an Item Box or balloon. Random Power Cores that are not of the leader's type are also rewarded after every 5 enemies destroyed. Enemies like the Egg Hammer, the Heavy Egg Hammer, and the E-2000R will give one random Power Core not of the leader's type for each one that is defeated. The last way to obtain them is via the checkpoints. Depending on which character is the leader, the appropriate Power Core will be given to that leader.
Power Cores also change the attacks of the characters, making them stronger. A fully leveled-up Speed character will be able to create tornadoes with Homing Attacks instead of having to use their Tornado Jump. A Fly character cannot destroy enemies with Thunder Shoot until they are at least to their second level. Upon the third level, they can grab Rings with Thunder Shoot. Power characters will get a more powerful Circular Attack with each level increase.
The Power Cores abilities are not permanent and will reset with each new level that's played. Also, if a life is lost during a level, the player will lose all of their level-ups and will have to collect the Power Cores all over again.