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