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