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