Here is the full list of all 76 Cave Story+ achievements.
Bad Ending
Obtained the Whimsical Star!
Obtained the Turbocharge!
Obtained the Machine Gun!
Obtained Chako's Lip Stick!
Beat by Toroko
For the rest of us!
Played Cave Story on April 29th!
Defeated Heavy Press!
Saved Curly Brace
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Found and collected all Life Capsules
Obtained the Arms Barrier!
Obtained the Iron Bond!
Obtained the Nemesis!
Returned Mister Little back home
Obtained Ma Pignon!
Found Mister Little!
Obtained the 290 Counter!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Obtained the Spur!
Obtained the Alien Medal!
Defeated Ironhead!
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Tow Rope!
Obtained the Snake!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Defeated Puu Black!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained Curly's Panties!
Defeated Curly...
Defeated Balfrog!
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!
Defeated G-Clone!
Defeated Ballos!
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
Spared Eggfish