As you take down Mavericks and solve problems, you'll gain experience points. These experience points are accumulated until you reach a certain threshold, at which point you level up.
Leveling in MHRPG involves not only levels, but Ranks as well. Your Rank determines how powerful of equipment you can use. You start off at Level 1, Rank E.
Every level you gain gives you an additional modification slot, with the exception of when you gain a new Rank. At Level 1, Rank E, you have a single modification slot. You can place this slot in whichever Part you wish, however, each Part can have a maximum of 3 slots.
Here's the breakdown of what you gain at each level.
Level 1: Rank E, Character Class Ability. 1 Modification Slot.
Level 2: 2 Slots
Level 3: 3 Slots
Level 4: Rank D Upgrade. Character Class Ability
Level 5: 4 Slots
Level 6: 5 Slots
Level 7: 6 Slots
Level 8: Rank C Upgrade. Character Class Ability
Level 9: 7 Slots
Level 10: 8 Slots
Level 11: 9 Slots
Level 12: Rank B Upgrade. Character Class Ability
Level 13: 10 Slots
Level 14: 11 Slots
Level 15: 12 Slots
Level 16: Rank A Upgrade. Character Class Ability
Level 17: 13 Slots
Level 18: 14 Slots
Level 19: 15 Slots
Level 20: Rank S Upgrade. Final Character Class Ability