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