Here is the full list of all 76 Cave Story+ achievements.
Defeated the Red Ogre!
Found and collected all Life Capsules
Obtained the Nemesis!
Returned Mister Little back home
Obtained the Iron Bond!
Obtained Ma Pignon!
Found Mister Little!
Obtained the 290 Counter!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Tow Rope!
Obtained the Arms Barrier!
Beat Omega with Jenka's Puppy
Defeated Balrog!
Bad Ending
Obtained the Bubbline!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
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 Curly's Panties!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Map System!
Obtained the Polar Star!
For the rest of us!
Obtained the Snake!
Obtained Chako's Lip Stick!
Obtained the Alien Medal!
Defeated Misery!
Defeated the Doctor!
Defeated the Muscle Doctor!
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
Beat by Toroko