Here is the full list of all 76 Cave Story+ achievements.
Saved Curly Brace
Spared Eggfish
Defeated the Red Ogre!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Bad Ending
Found and collected all Missile upgrades
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Tow Rope!
Obtained the Snake!
Beat by Toroko
Beat Omega with Jenka's Puppy
Defeated Omega!
Obtained Curly's Panties!
Obtained Chako's Lip Stick!
Obtained the Bubbline!
Obtained the Super Missile Launcher!
Obtained the Machine Gun!
Obtained the Blade!
Obtained the Life Pot!
Obtained the Arms Barrier!
Defeated Curly...
Defeated Toroko...
Defeated Puu Black!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Defeated Ironhead!
Defeated the Twin Dragons!
Found and collected all Life Capsules
Accumulated XX,XXX of damage
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated Balfrog!
Obtained the Polar Star!
Obtained the Fireball!
Obtained the Missile Launcher!
Obtained the Map System!
Defeated Balrog!
Defeated Igor!
Defeated Balrog, again...
For the rest of us!
Obtained the Alien Medal!
Obtained the 290 Counter!
Obtained the Turbocharge!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
Mask Mimiga Ending
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
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!