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