Here is the full list of all 76 Cave Story+ achievements.
Completed Boss Attack in Less Than 8:00
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Wind Fortress in Less Than 4:00
Completed Blood Stained Sanctuary in Less Than 4:00
Completed Boss Attack in Less Than 10:00
Completed Blood Stained Sanctuary in Less Than 5:00
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Boss Attack in Less Than 12:00
Completed Boss Attack in Less Than 15:00
No Life Capsules (Best Ending)
No Life Capsules (Good Ending)
Mask Mimiga Ending
Bad Ending
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Completed Boss Attack
Completed Blood Stained Sanctuary
Completed Blood Stained Sanctuary in Less Than 6:00
Found and collected all Life Capsules
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Alien Medal!
Spared Eggfish
Completed Hard Mode
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained Curly's Panties!
Obtained the Tow Rope!
Obtained the Snake!
Obtained Chako's Lip Stick!
Beat by Toroko
Played Cave Story on April 29th!
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Defeated the Twin Dragons!
Found and collected all Missile upgrades
Defeated Ironhead!
Defeated the Core!
Accumulated XX,XXX of damage
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!
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Obtained the Bubbline!
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
For the rest of us!