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