Here is the full list of all 76 Cave Story+ achievements.
Mask Mimiga Ending
Obtained the Whimsical Star!
Obtained the Spur!
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
Completed Boss Attack in Less Than 15:00
Completed Blood Stained Sanctuary
Spared Eggfish
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Nemesis!
Returned Mister Little back home
Obtained the Iron Bond!
Obtained Ma Pignon!
Found Mister Little!
Found and collected all Life Capsules
Obtained the 290 Counter!
Found and collected all Missile upgrades
Obtained the Alien Medal!
Obtained the Tow Rope!
Obtained the Snake!
Obtained Curly's Panties!
Obtained Chako's Lip Stick!
Beat by Toroko
Played Cave Story on April 29th!
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated Misery!
Defeated the Doctor!
Bad Ending
Defeated the Twin Dragons!
Obtained the Bubbline!
Obtained the Map System!
Defeated Ironhead!
Defeated the Core!
Obtained the Arms Barrier!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Accumulated XX,XXX of damage
Defeated Puu Black!
Obtained the Turbocharge!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
For the rest of us!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Polar Star!
Defeated G-Clone!
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Hard Mode
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 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