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