move contents from one file to another based on line number or content

I want a script that will move everything beyond a certain line number or beyond a certain content word into another file.
For example, if file A has this:
first line
second line
third line
forth line
fifth line
sixth line

I want to run a script that will move everything beyond the third line to the END of file B, OR it would be even better if the script will move everything beyond the specific words "third line" into file B. That way I could just put in some random characters where I want the cutoff to be and never have to worry about the line number. I am not very good at scripting and do not know where to start. Any help would be greatly appreciated.

What work have you started on this script? This forum is intended to help you resolve issues in your solutions, not necessarily as a consultancy to do the work for you. This is not to say you won't get a solution, but you really should try it on your own first.

awk ' ok==1 {print $0} 
       /third line/ {ok=1} ' inputfile > outputfile

substitute the words "thrid line" with your random characters. NOTE: printing starts with the LINE AFTER the file record with "third line" in it.

Try

awk '/third line/,0' FileA | awk 'NR>1' > FileB

The green marked part is optional, if you run the code without it, then the search string "third line" will appear in FileB as well.

Btw. I agree with dunkar70.

Actually it looks somewhat like homework, but not enough to warrant moderation.