During the main game talk to Silvia to request a friendly with teams that you have defeated in the regional and national Football Frontier Tournament. The final team for each of those routes is unlocked after beating the story mode: Royal Academy for the regional qualifiers, Zeus Academy for the national competition. When you defeat each of those teams you are given a rank based on the goal difference you obtained: 'S' rank for 5 or more goals, 'A' rank for 4 goals and 'B' rank for everything else. Defeating a team allows you to challenge the next one and at the end of each route there is a chest containing two rewards: one for clearing the route, another for clearing the route with an 'S' rank on every match. The 'S' Rank rewards are manuals for a powerful shot and keeper skill that you cannot get anywhere else in the game.
Unlock Inazuma Boots: Defeat every team from the national competition (bottom) route.
Unlock Winged Boots: Defeat every team from the regional qualifier (top) route.
Unlock Inazuma-1 Drop: Defeat every team from the national competition (bottom) route with an 'S' rank.
Unlock Clone Block: Defeat every team from the regional qualifier (top) route with an 'S' rank.
When you have completed the main story talk to Raimon at the headmaster's office to unlock two new competition routes. Then complete the indicated task in the new routes to be rewarded with the corresponding bonus. An 'S' rank is obtained by winning with a 5 goal difference.
Unlock Gold Power Pendant: Complete the lower route.
Unlock Gold Shield Pendant: Complete the upper route.
Unlock Dual Smash: Defeat every team on the lower route with an 'S' rank.
Unlock Dark Phoenix: Defeat every team on the upper route with an 'S' rank.