Here is the full list of all 76 Cave Story+ achievements.
Obtained the Turbocharge!
Obtained the Machine Gun!
Beat by Toroko
Mask Mimiga Ending
Played Cave Story on April 29th!
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Normal Ending
Completed Easy Mode
Spared Eggfish
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Found and collected all Life Capsules
Obtained the Nemesis!
Returned Mister Little back home
Obtained the 290 Counter!
Obtained the Iron Bond!
Obtained Ma Pignon!
Found Mister Little!
Bad Ending
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Alien Medal!
Defeated Ironhead!
Accumulated XX,XXX of damage
Defeated the Core!
Obtained the Tow Rope!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Defeated Puu Black!
Obtained the Arms Barrier!
Obtained the Snake!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained Curly's Panties!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Bubbline!
Obtained Chako's Lip Stick!
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Map System!
Defeated Balrog!
Obtained the Polar Star!
Defeated G-Clone!
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
For the rest of us!