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