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