How worried should I be about bleeding along my gums?

Check it out. You should be worried enough that you check it out. Most likely, it is some form of local disease, i.e., gingivitis. But this is not trivial. It could reflect more systemic bleeding problems. Either way, you need a diagnosis, and probably intervention. If there are no other signs of easy bleeding, start with a good dental evaluation, then if more answers are needed, see a good internist.