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