Here is the full list of all 76 Cave Story+ achievements.
Obtained the Spur!
Obtained the Whimsical Star!
Bad Ending
Obtained the Snake!
Obtained the Blade!
Obtained the Nemesis!
Found Mister Little!
Defeated Toroko...
Defeated Puu Black!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Defeated Ironhead!
Defeated G-Clone!
Completed Wind Fortress in Less Than 7:00
Returned Mister Little back home
Played Cave Story on April 29th!
Beat by Toroko
Completed Easy Mode
Defeated Ballos!
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!
Obtained the Tow Rope!
Obtained the Arms Barrier!
Defeated Heavy Press!
Obtained the Super Missile Launcher!
Found and collected all Missile upgrades
Saved Curly Brace
Found and collected all Life Capsules
Normal Ending
Spared Eggfish
Obtained the Bubbline!
Obtained the Life Pot!
Obtained Curly's Panties!
Defeated Balrog, again...
Defeated Balfrog!
Defeated Curly...
Defeated Omega!
Beat Omega with Jenka's Puppy
Accumulated XX,XXX of damage
Defeated Igor!
Obtained Chako's Lip Stick!
Obtained the Fireball!
For the rest of us!
Obtained the Polar Star!
Obtained the Missile Launcher!
Obtained the Map System!
Defeated Balrog!
Obtained the Machine Gun!
Obtained the Alien Medal!
Obtained the Turbocharge!
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