Here is the full list of all 76 Cave Story+ achievements.
Obtained Curly's Panties!
Defeated Heavy Press!
Saved Curly Brace
Normal Ending
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Obtained the Iron Bond!
Obtained Ma Pignon!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Spur!
Defeated Ironhead!
Defeated the Core!
Obtained the Super Missile Launcher!
Obtained the Tow Rope!
Defeated Monster X!
Defeated Balrog, yet again...
Beat Omega with Jenka's Puppy
Obtained the Bubbline!
Obtained the Missile Launcher!
Obtained the Machine Gun!
Obtained the Blade!
Obtained the Life Pot!
Obtained the Arms Barrier!
Obtained the Turbocharge!
Defeated Curly...
Defeated Omega!
Defeated Toroko...
Defeated Puu Black!
Accumulated XX,XXX of damage
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
Obtained the Snake!
Obtained the Nemesis!
Obtained Chako's Lip Stick!
Obtained the Whimsical Star!
Obtained the Alien Medal!
Obtained the 290 Counter!
Found Mister Little!
Defeated G-Clone!
Defeated Ballos!
Bad Ending
Mask Mimiga Ending
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Easy Mode
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
Returned Mister Little back home
Found and collected all Life Capsules
Beat by Toroko
For the rest of us!