Here is the full list of all 76 Cave Story+ achievements.
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Boss Attack in Less Than 8:00
Completed Wind Fortress in Less Than 4:00
Played Cave Story on April 29th!
Obtained the Alien Medal!
Obtained the Turbocharge!
Beat Omega with Jenka's Puppy
Obtained Curly's Panties!
Obtained the Machine Gun!
Obtained Chako's Lip Stick!
Beat by Toroko
Completed Wind Fortress in Less Than 5:00
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
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 Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Mask Mimiga Ending
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Found and collected all Missile upgrades
Completed Easy Mode
Normal Ending
Completed Hard Mode
Spared Eggfish
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Bad Ending
Defeated the Twin Dragons!
Obtained the Whimsical Star!
Obtained the Bubbline!
Obtained the Spur!
Defeated Ironhead!
Obtained the Tow Rope!
Defeated the Core!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Obtained the Snake!
Obtained the Arms Barrier!
Accumulated XX,XXX of damage
Defeated Puu Black!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Defeated Curly...
Defeated Omega!
Found and collected all Life Capsules
Defeated Balfrog!
Defeated Balrog, again...
For the rest of us!
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!