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