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