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