Here is the full list of all 76 Cave Story+ achievements.
Played Cave Story on April 29th!
Mask Mimiga Ending
Defeated Ballos!
Spared Eggfish
Defeated Heavy Press!
Saved Curly Brace
Completed Blood Stained Sanctuary
Normal Ending
Completed Easy Mode
Obtained the Snake!
Obtained the Iron Bond!
Defeated the Undead Core!
Returned Mister Little back home
Obtained the Whimsical Star!
Defeated the Red Ogre!
Obtained the Nemesis!
Defeated the Muscle Doctor!
Obtained the Tow Rope!
Found Mister Little!
Obtained the 290 Counter!
Obtained Ma Pignon!
Found and collected all Life Capsules
Defeated the Doctor!
Defeated Misery!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Bubbline!
Defeated Ironhead!
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Turbocharge!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Arms Barrier!
Defeated Puu Black!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained Curly's Panties!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
For the rest of us!
Obtained the Spur!
Obtained Chako's Lip Stick!
Obtained the Alien Medal!
Defeated G-Clone!
Bad Ending
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Hard Mode
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Blood Stained Sanctuary in Less Than 4:00
Completed Blood Stained Sanctuary in Less Than 5:00
Completed Blood Stained Sanctuary in Less Than 6:00
Completed Boss Attack
Completed Boss Attack in Less Than 8:00
Completed Boss Attack in Less Than 10:00
Completed Boss Attack in Less Than 12:00
Completed Boss Attack in Less Than 15:00
Completed Wind Fortress in Less Than 4:00
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Beat by Toroko