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