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