Here is the full list of all 34 PEG achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Obtain all achievements.
Clear 30 daily levels total.
Clear 20 daily levels total.
Clear 10 daily levels total.
Clear all CROSS levels.
Clear a CROSS level.
Clear all HEXAGON levels.
Unlock the CROSS board.
Clear a daily level.
Unlock daily levels.
Move 3000 PEG pieces.
Clear an HEXAGON level.
Clear all DOUBLE DIAMOND levels.
Unlock the HEXAGON board.
Move 2500 PEG pieces.
Move 2000 PEG pieces.
Clear a DOUBLE DIAMOND level.
Clear all CIRCLE levels.
Unlock the DOUBLE DIAMOND board.
Move 1500 PEG pieces.
Move 1000 PEG pieces.
Beat the 7th CIRCLE level while only moving the red PEG piece ONCE.
Clear a CIRCLE level.
Clear all SQUARE board levels.
Unlock the CIRCLE board.
Move 500 PEG pieces.
Clear a SQUARE level.
Clear all STAR board levels.
Clear a STAR board level.
Unlock the SQUARE board.
Clear all TRIANGLE board levels.
Unlock the STAR board.
Clear a TRIANGLE level.
Complete the tutorial.