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 3000 PEG pieces.
Obtain all achievements.
Clear all CROSS levels.
Clear 30 daily levels total.
Move 2500 PEG pieces.
Clear 20 daily levels total.
Move 2000 PEG pieces.
Clear a CROSS level.
Clear 10 daily levels total.
Clear all HEXAGON levels.
Unlock the CROSS board.
Move 1500 PEG pieces.
Clear a daily level.
Unlock daily levels.
Clear an HEXAGON level.
Clear all DOUBLE DIAMOND levels.
Unlock the HEXAGON board.
Move 1000 PEG pieces.
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.
Unlock the STAR board.
Clear all TRIANGLE board levels.
Clear a TRIANGLE level.
Complete the tutorial.