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