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