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