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