How can I quickly delete a line in VIM starting at the cursor position?

Use D. See docs for further information.

(Edited to include commenter's good additions:)

D or its equivalent d$ will delete the rest of the line and leave you in command mode. C or c$ will delete the rest of the line and put you in insert mode, and new text will be appended to the line.

This is part of vitutor and vimtutor, excellent "reads" for vim beginners.