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