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