Here is the full list of all 76 Cave Story+ achievements.
Completed Blood Stained Sanctuary
Completed Blood Stained Sanctuary in Less Than 5:00
Completed Blood Stained Sanctuary in Less Than 6:00
Completed Boss Attack
No Life Capsules (Good Ending)
Obtained the Alien Medal!
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
Defeated G-Clone!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Mask Mimiga Ending
Obtained Curly's Panties!
Obtained Chako's Lip Stick!
Played Cave Story on April 29th!
Defeated Ballos!
Spared Eggfish
Defeated Heavy Press!
Saved Curly Brace
Found and collected all Life Capsules
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the Bubbline!
Obtained the Whimsical Star!
Obtained the Spur!
Beat by Toroko
Completed Easy Mode
Bad Ending
Found and collected all Missile upgrades
Obtained the Snake!
Defeated the Undead Core!
Normal Ending
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Obtained the 290 Counter!
Defeated the Twin Dragons!
Defeated Ironhead!
Obtained the Tow Rope!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Missile Launcher!
Obtained the Machine Gun!
Obtained the Blade!
Obtained the Life Pot!
Obtained the Arms Barrier!
Obtained the Turbocharge!
Defeated Curly...
Defeated Omega!
Defeated Toroko...
Defeated Puu Black!
Beat Omega with Jenka's Puppy
Accumulated XX,XXX of damage
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Obtained the Map System!
Defeated Igor!
Defeated Balrog!
Obtained the Polar Star!
For the rest of us!
No Life Capsules (Best Ending)
Completed Hard Mode
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Blood Stained Sanctuary in Less Than 4:00
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