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