Here is the full list of all 76 Cave Story+ achievements.
Completed Blood Stained Sanctuary in Less Than 3:00
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
Defeated G-Clone!
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 Blood Stained Sanctuary
No Life Capsules (Best Ending)
Defeated Ballos!
Defeated Heavy Press!
Mask Mimiga Ending
No Life Capsules (Good Ending)
Completed Hard Mode
Spared Eggfish
Obtained the Alien Medal!
For the rest of us!
Played Cave Story on April 29th!
Beat by Toroko
Defeated the Red Ogre!
Saved Curly Brace
Found and collected all Life Capsules
Obtained the Nemesis!
Returned Mister Little back home
Obtained the Iron Bond!
Obtained Ma Pignon!
Found Mister Little!
Obtained the Snake!
Obtained the Spur!
Obtained the Tow Rope!
Obtained Curly's Panties!
Obtained the Whimsical Star!
Obtained the 290 Counter!
Bad Ending
Obtained the Polar Star!
Obtained the Fireball!
Obtained the Bubbline!
Obtained the Missile Launcher!
Obtained the Super Missile Launcher!
Obtained the Machine Gun!
Obtained the Blade!
Obtained the Map System!
Obtained Chako's Lip Stick!
Obtained the Life Pot!
Obtained the Arms Barrier!
Obtained the Turbocharge!
Defeated Balrog!
Defeated Igor!
Defeated Balrog, again...
Defeated Balfrog!
Defeated Curly...
Defeated Omega!
Defeated Toroko...
Defeated Puu Black!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Defeated Ironhead!
Defeated the Twin Dragons!
Defeated Misery!
Defeated the Doctor!
Defeated the Muscle Doctor!
Defeated the Undead Core!
Normal Ending
Completed Easy Mode
Beat Omega with Jenka's Puppy
Found and collected all Missile upgrades
Accumulated XX,XXX of damage