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