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