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