Here is the full list of all 76 Cave Story+ achievements.
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
Found and collected all Life Capsules
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Tow Rope!
Obtained Chako's Lip Stick!
Beat by Toroko
Obtained the Snake!
Obtained Curly's Panties!
Defeated Balrog!
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Defeated Ironhead!
Defeated the Core!
Obtained the Turbocharge!
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 Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Obtained the Bubbline!
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Map System!
Obtained the Polar Star!
Obtained the Alien Medal!
Defeated G-Clone!
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
For the rest of us!