To unlock more puzzle-stages, simply beat the ten puzzles before that ones. When you unlock more puzzles, you automatically get the same amount of Saves for your own puzzles.
Unlock the credits by either solving all 100 puzzles, or by completing Challenge mode.
Beat the first rank time record in Easy and Hard modes to unlock Sudden Death mode at the Time Attack menu.
Complete all puzzles 100% in Polarium Reference to unlock three Special puzzles.