Here is the full list of all 76 Cave Story+ achievements.
Completed Wind Fortress in Less Than 4:00
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 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
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
Spared Eggfish
Obtained the Turbocharge!
Beat Omega with Jenka's Puppy
Beat by Toroko
Played Cave Story on April 29th!
For the rest of us!
Obtained the Spur!
Obtained the Nemesis!
Obtained the Whimsical Star!
Obtained the Arms Barrier!
Obtained the Alien Medal!
Obtained the 290 Counter!
Found Mister Little!
Obtained Ma Pignon!
Obtained the Iron Bond!
Defeated Ironhead!
Defeated the Twin Dragons!
Defeated the Red Ogre!
Defeated Misery!
Defeated the Doctor!
Defeated the Muscle Doctor!
Defeated the Undead Core!
Defeated Heavy Press!
Defeated Ballos!
Bad Ending
Normal Ending
Mask Mimiga Ending
Completed Easy Mode
Saved Curly Brace
Found and collected all Missile upgrades
Returned Mister Little back home
Found and collected all Life Capsules
Obtained the Bubbline!
Obtained the Super Missile Launcher!
Obtained the Machine Gun!
Obtained the Snake!
Obtained the Blade!
Obtained the Life Pot!
Obtained the Tow Rope!
Obtained Curly's Panties!
Defeated Balrog, again...
Defeated Balfrog!
Defeated Curly...
Defeated Omega!
Defeated Toroko...
Defeated Puu Black!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Fireball!
Obtained Chako's Lip Stick!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!