Upon finishing Chapter 13, when you go to do the Free Mission at Shinjuku, beat it and you will unlock the Jet. For Sky Missions, on the Dragon Select Menu, you should normally see Red Dragon Level 1, 2, 3 and Chaos Dragon. If you go one below Chaos Dragon{which means while you highlighting Chaos Dragon, just press down}, and you will be highlighting '???', select it and you will hear Manah's voices. After that, get on to the mission and you should be riding on a Jet. The Jet looks like, a Jet, unlike the Dragons you have been using and it's only available for Sky Missions.
To level up weak weapons quickly, go to story mode, chapter 1, verse 2 (to the castle) on easy mode. Equip Spiked Wisdom, Butcher's Joy, or another weapon that's magic creates an aura around you that can quickly kill enemies. Charge up the magic meter, cast the spell, then quickly press/hold R2, switch weapons to the one you want to level up, and then run around at enemies. The aura will kill enemies for as long as it lasts while the kills go toward the currently equipped weapon. Once the aura runs out, just switch back and repeat.
During any ground mission if you see enemies on your radar on the other side of the wall, you can begin your run attack until you see the flash that you are going full speed. Then press the attack button to send your attack through the wall, killing the enemies on the other side.