Here is the full list of all 76 Cave Story+ achievements.
Played Cave Story on April 29th!
For the rest of us!
Completed Blood Stained Sanctuary in Less Than 6:00
Completed Boss Attack in Less Than 8:00
Completed Hard Mode
Completed Boss Attack in Less Than 10:00
No Life Capsules (Good Ending)
Completed Wind Fortress in Less Than 4:00
Completed Boss Attack in Less Than 12:00
Completed Boss Attack in Less Than 15:00
Completed Blood Stained Sanctuary
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!
Completed Boss Attack
Defeated Ballos!
Defeated Heavy Press!
Mask Mimiga Ending
Completed Easy Mode
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained the Alien Medal!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Tow Rope!
Beat Omega with Jenka's Puppy
Normal Ending
Spared Eggfish
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Found and collected all Life Capsules
Obtained Curly's Panties!
Obtained the Snake!
Bad Ending
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Bubbline!
Defeated Ironhead!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Turbocharge!
Defeated Puu Black!
Obtained the Arms Barrier!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Accumulated XX,XXX of damage
Obtained Chako's Lip Stick!
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Beat by Toroko
Obtained the Map System!
Obtained the Polar Star!
No Life Capsules (Best Ending)
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