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