Here is the full list of all 34 PEG achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Move 1000 PEG pieces.
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.
Clear a DOUBLE DIAMOND level.
Clear all DOUBLE DIAMOND levels.
Unlock the HEXAGON board.
Clear an HEXAGON level.
Clear all HEXAGON levels.
Unlock the CROSS board.
Clear a CROSS level.
Clear all CROSS levels.
Move 1500 PEG pieces.
Move 2000 PEG pieces.
Move 2500 PEG pieces.
Move 3000 PEG pieces.
Unlock daily levels.
Clear a daily level.
Clear 10 daily levels total.
Clear 20 daily levels total.
Clear 30 daily levels total.
Obtain all achievements.