Here is the full list of all 23 Core Rescue achievements.
Defeat the Firewall
Defeat the Carrier
Defeat the Defenders
Defeat Eric
Defeat the Trojan
Defeat the Core
Unlock all Ships
Defeat 100 Viruses
Defeat 1000 Viruses
Defeat 10000 Viruses
Dash Through 10 Enemies
Dash through 100 Enemies
Die 100 Times
Win on Easy Mode
Win on Medium Mode
Win on Hard Mode
Use 100 special abilities
Unlock the Arrow
Unlock the Defender
Unlock the A20
Unlock the stealth bomber
Unlocked the V.I.R.S.
Get 1 million points