Here is the full list of all 76 Cave Story+ achievements.
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Played Cave Story on April 29th!
For the rest of us!
Spared Eggfish
Completed Easy Mode
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
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!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Alien Medal!
Obtained the Snake!
Obtained the Tow Rope!
Beat Omega with Jenka's Puppy
Obtained Curly's Panties!
Obtained the Bubbline!
Obtained Chako's Lip Stick!
Beat by Toroko
Normal Ending
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Bad Ending
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Defeated Ironhead!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Arms Barrier!
Defeated Puu Black!
Obtained the Turbocharge!
Obtained the Blade!
Defeated Toroko...
Accumulated XX,XXX of damage
Obtained the Life Pot!
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Obtained the Map System!
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Polar Star!
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