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