"I forgot to take today's notes."
"I forgot to take notes today" is a better translation.
"take notes" specifically means "to write a summary" (where take means "to write" and notes means "a summary")
Although the Chinese sentence states "…写今天的笔记" litterally "…write today's notes", the suggested translation "…take today's notes" breaks the phrase "take notes".
This causes the words revert to their individual meanings (i.e. "take" is no longer understood to mean "to write" and reverts to "to lay hold, carry, bring, or recieve, etc") and sounds as if the speaker forgot to pick up a classroom handout from the instructor rather than he forgot to write his own personal notes during class.
Alternatively, to preserve both the Chinese sentence structure and meaning you could use "write" "write down" or "take down" today's notes.
However, these options are less direct and less natural than "take notes"