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