Here is the full list of all 76 Cave Story+ achievements.
Saved Curly Brace
Found and collected all Life Capsules
Normal Ending
Completed Easy Mode
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 Bubbline!
Defeated Ironhead!
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Tow Rope!
Obtained the Super Missile Launcher!
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!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Polar Star!
Obtained the Fireball!
Obtained the Missile Launcher!
Obtained the Map System!
Defeated Igor!
Obtained the Snake!
Obtained the Spur!
Obtained the Nemesis!
Obtained Chako's Lip Stick!
Obtained Curly's Panties!
Obtained the Whimsical Star!
Obtained the Alien Medal!
Found Mister Little!
Defeated Balrog!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
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
Returned Mister Little back home
Beat by Toroko
For the rest of us!