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 all CROSS levels.
Clear all HEXAGON levels.
Clear a CROSS level.
Unlock the CROSS board.
Clear 30 daily levels total.
Clear 20 daily levels total.
Clear all DOUBLE DIAMOND levels.
Clear 10 daily levels total.
Move 3000 PEG pieces.
Unlock daily levels.
Clear a daily level.
Move 2500 PEG pieces.
Clear all CIRCLE levels.
Clear an HEXAGON level.
Unlock the HEXAGON board.
Move 2000 PEG pieces.
Clear a DOUBLE DIAMOND level.
Unlock the DOUBLE DIAMOND board.
Move 1500 PEG pieces.
Move 1000 PEG pieces.
Clear all SQUARE board levels.
Clear all STAR board levels.
Clear a CIRCLE level.
Beat the 7th CIRCLE level while only moving the red PEG piece ONCE.
Unlock the CIRCLE board.
Move 500 PEG pieces.
Clear a SQUARE level.
Unlock the SQUARE board.
Clear a STAR board level.
Unlock the STAR board.
Clear all TRIANGLE board levels.
Clear a TRIANGLE level.
Complete the tutorial.