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