Cally can double jump (and eventually triple jump) to collect out of reach gems and avoid obstacles such as water and spike pits. Try to time your first jump just when Cally is coming off the tip of a ledge to get the best speed and distance. To clear the more difficult gaps, you will need to use a second jump while in the air.
Weapons
Cally will be given a random weapon (shotgun, assault rifle, flamethrower and others) that will never run out of ammunition. Because some enemies require a lot of shots to kill, they must be taken out at a distance. The longer you use a weapon and the more gems you collect, the faster it will level up. This will increase its damage capacity when fired at enemies. The shotgun is a good choice, as it has good range and damage that spreads out to hit enemies both above and below you. The boomerang is also a good choice, because it hits enemies from the front and behind, and you can throw many of them at once. Use the sword is for up-close melee attacks. It should be a last resort to hit enemies that have gotten too close. When used against larger enemies, make sure to avoid their attack range. You can use the money dropped by enemies to upgrade the sword at the shop. The more enemies you defeat, the faster you can upgrade your equipment.