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