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