Here is the full list of all 76 Cave Story+ achievements.
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Completed Boss Attack
Completed Boss Attack in Less Than 15:00
Defeated Ballos!
Defeated Heavy Press!
Mask Mimiga Ending
Spared Eggfish
Obtained the Snake!
Beat Omega with Jenka's Puppy
Beat by Toroko
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Nemesis!
Returned Mister Little back home
Obtained the Iron Bond!
Obtained Ma Pignon!
Found Mister Little!
Found and collected all Life Capsules
Obtained the 290 Counter!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Alien Medal!
Obtained the Tow Rope!
Obtained Curly's Panties!
Obtained the Bubbline!
Obtained Chako's Lip Stick!
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Bad Ending
Defeated the Twin Dragons!
Accumulated XX,XXX of damage
Found and collected all Missile upgrades
Defeated Ironhead!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Arms Barrier!
Defeated Puu Black!
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!
Obtained the Map System!
Defeated Balrog!
Obtained the Polar Star!
For the rest of us!
Obtained the Turbocharge!
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
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 in Less Than 8:00
Completed Boss Attack in Less Than 10:00
Completed Boss Attack in Less Than 12:00
Completed Wind Fortress in Less Than 4:00
Played Cave Story on April 29th!