Here is the full list of all 76 Cave Story+ achievements.
Obtained the Snake!
Beat Omega with Jenka's Puppy
Obtained Curly's Panties!
Obtained Chako's Lip Stick!
Defeated Balrog!
Beat by Toroko
Played Cave Story on April 29th!
Normal Ending
Completed Easy Mode
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Bad Ending
Defeated the Twin Dragons!
For the rest of us!
Found and collected all Missile upgrades
Defeated Ironhead!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Arms Barrier!
Defeated Puu Black!
Accumulated XX,XXX of damage
Obtained the Turbocharge!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Obtained the Bubbline!
Defeated Balrog, again...
Defeated Balfrog!
Obtained the Fireball!
Obtained the Map System!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Polar Star!
Obtained the Spur!
Obtained the Nemesis!
Obtained the Tow Rope!
Obtained the Whimsical Star!
Obtained the Alien Medal!
Obtained the 290 Counter!
Found Mister Little!
Obtained Ma Pignon!
Obtained the Iron Bond!
Defeated the Red Ogre!
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
Spared Eggfish
Saved Curly Brace
Returned Mister Little back home
Found and collected all Life Capsules