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