Can I just get a new screen and is it not covered under AppleCare?
If your screen is cracked and it is showing "lots" of colors, it means your digitizer is cracked/broken as well.
AFIK, AppleCare doesn't cover accidental damage, AppleCare+ does.
If you don't have AppleCare+, then you need to take it in to an Apple Store or to an authorized repair shop.
All of this is spelled out clearly at Apple's service page for iPhones:
For an out of warranty screen repair in the US you should expect to pay $129 and if more things are broken than just the screen the total could go up to $269 before it's considered "totaled" or "beyond economical repair". Apple support should be able to tell you if your phone has AppleCare plus from the second link above for iPhone service center information.