Here is the full list of all 76 Cave Story+ achievements.
Saved Curly Brace
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 Whimsical Star!
Found Mister Little!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Spur!
Obtained the Alien Medal!
Defeated Ironhead!
Obtained the Tow Rope!
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Arms Barrier!
Defeated Puu Black!
Obtained the Snake!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Defeated Omega!
Obtained Curly's Panties!
Defeated Curly...
Obtained the Bubbline!
Defeated Balfrog!
Defeated Balrog, again...
Obtained Chako's Lip Stick!
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
Obtained the Machine Gun!
Obtained the Turbocharge!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
Bad Ending
Normal Ending
Mask Mimiga Ending
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Easy Mode
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
Beat Omega with Jenka's Puppy
Beat by Toroko
For the rest of us!