Here is the full list of all 76 Cave Story+ achievements.
Played Cave Story on April 29th!
Completed Boss Attack in Less Than 10:00
No Life Capsules (Best Ending)
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Boss Attack in Less Than 12:00
Completed Blood Stained Sanctuary
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
For the rest of us!
Completed Boss Attack in Less Than 15:00
Completed Boss Attack
No Life Capsules (Good Ending)
Completed Hard Mode
Obtained the Alien Medal!
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
Defeated G-Clone!
Mask Mimiga Ending
Spared Eggfish
Obtained Chako's Lip Stick!
Completed Easy Mode
Defeated Ballos!
Defeated Heavy Press!
Beat by Toroko
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Whimsical Star!
Obtained the Nemesis!
Returned Mister Little back home
Obtained the Iron Bond!
Found Mister Little!
Obtained Ma Pignon!
Obtained the 290 Counter!
Bad Ending
Obtained Curly's Panties!
Obtained the Spur!
Obtained the Tow Rope!
Obtained the Snake!
Found and collected all Life Capsules
Normal Ending
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
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!
Accumulated XX,XXX of damage
Defeated Puu Black!
Obtained the Arms Barrier!
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!
Obtained the Bubbline!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
Completed Boss Attack in Less Than 8:00