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