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.
Move 2500 PEG pieces.
Move 2000 PEG pieces.
Clear a CROSS level.
Clear all HEXAGON levels.
Unlock the CROSS board.
Clear 30 daily levels total.
Clear 20 daily levels total.
Move 1500 PEG pieces.
Clear 10 daily levels total.
Move 1000 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 500 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.
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.