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