Here is the full list of all 76 Cave Story+ achievements.
Played Cave Story on April 29th!
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Blood Stained Sanctuary
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 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 Boss Attack
Completed Hard Mode
No Life Capsules (Best Ending)
Normal Ending
No Life Capsules (Good Ending)
Completed Easy Mode
Spared Eggfish
Obtained the Turbocharge!
Obtained Chako's Lip Stick!
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
Defeated G-Clone!
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
Mask Mimiga Ending
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Found and collected all Life Capsules
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Bad Ending
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Arms Barrier!
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!
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!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Obtained the Missile Launcher!
Defeated Balrog, again...
Obtained the Fireball!
Obtained the Bubbline!
Defeated Igor!
Defeated Balrog!
Beat by Toroko
Obtained the Map System!
Obtained the Polar Star!
For the rest of us!