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