Here is the full list of all 76 Cave Story+ achievements.
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Blood Stained Sanctuary in Less Than 4:00
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
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
Completed Boss Attack in Less Than 15:00
Completed Blood Stained Sanctuary
Completed Blood Stained Sanctuary in Less Than 5:00
Completed Blood Stained Sanctuary in Less Than 6:00
Completed Wind Fortress in Less Than 4:00
Mask Mimiga Ending
Obtained Chako's Lip Stick!
For the rest of us!
Completed Hard Mode
Played Cave Story on April 29th!
Beat by Toroko
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Obtained Curly's Panties!
Completed Easy Mode
Obtained the Snake!
Defeated G-Clone!
Spared Eggfish
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Defeated the Red Ogre!
Obtained the Iron Bond!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Obtained Ma Pignon!
Found and collected all Life Capsules
Obtained the 290 Counter!
Found and collected all Missile upgrades
Obtained the Bubbline!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Tow Rope!
Beat Omega with Jenka's Puppy
Normal Ending
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Accumulated XX,XXX of damage
Bad Ending
Defeated the Twin Dragons!
Obtained the Alien Medal!
Defeated Ironhead!
Defeated the Core!
Obtained the Turbocharge!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Defeated Puu Black!
Obtained the Arms Barrier!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Defeated Omega!
Obtained the Machine Gun!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Obtained the Map System!
Defeated Balrog!
Obtained the Polar Star!