You don't need to get gems dashless for the achievements, so if there appears to be an easier way to get the gem than is shown in the text/video guide, just go for it!
At first you won't be able to do the levels in order due to how the world map works. Once you get the upgrades for destroying obstacles and placing tiles (can be found in shops), you can freely explore the world map.
Dashing at bosses is an effective way to deal damage.
In order to be able to fight the true last boss, you must collect all 7 rainbow crystal fragments (one at each pond), which will grant you the 'Rainbow Crystal' item. Once you have the item, fight the final boss as normal until the last phase; you will be able to damage the crystal instead. Once you've broken the crystal the true final boss appears.
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