Here is the full list of all 76 Cave Story+ achievements.
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Found and collected all Life Capsules
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Bad Ending
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Defeated Ironhead!
Obtained the Super Missile Launcher!
Obtained the Arms Barrier!
Obtained the Turbocharge!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Defeated Puu Black!
Obtained the Blade!
Defeated Toroko...
Accumulated XX,XXX of damage
Obtained the Life Pot!
Obtained Curly's Panties!
Obtained the Machine Gun!
Defeated Omega!
Defeated Curly...
Defeated Balfrog!
Obtained the Bubbline!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Map System!
Obtained the Polar Star!
Obtained the Snake!
Obtained the Spur!
Obtained Chako's Lip Stick!
Obtained the Tow Rope!
Obtained the Whimsical Star!
Obtained the Alien Medal!
Obtained the 290 Counter!
Obtained Ma Pignon!
Obtained the Iron Bond!
Defeated Balrog!
Defeated the Red Ogre!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
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
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Played Cave Story on April 29th!
Spared Eggfish
Beat Omega with Jenka's Puppy
Saved Curly Brace
Beat by Toroko
For the rest of us!