## Ridiculous script needs....

This is probably really simple, and the reasoning is ridiculous, but I am a scripting noob, and have only a small clue...
Let's say I have 10 text files, each with one line of text. I want to cat each file, take that one line of text, and append it to a separate file (i.e. if I have files 1-10.txt, I want to end up with a file 'links.txt' that has ten lines of text, one from each of the ten files). This is ridiculous...
I then needed to take each line from links.txt and open them in firefox under new tabs. I have this part done. I tested it and all is well (nice time to learn a loop!), but I am having one hell of a time figuring out how to take 10 files and put their info into one file. I tried 'cat *.txt >> links.txt' but that basically puts all of the info in paragraph form, where I need something that will do one line of text, then do a newline, then add a line of text, newline, and so forth.