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