I'd assume that your joists would be running perpendicular to both the direction your hardwood is running, as well as the direction that the wood on the basement ceiling is run. That's obviously not an absolute as people do odd things, but just from seeing the hardwood/ceiling, that's what I would look for first.
Best way to tell might be to take the two grills off and stick your head in for a peak. Might as well in any event; the more you know about how they would have framed that all in, the better (assuming it's just air through framing void and isn't ducted somehow). And how your joists run will determine how you can run whatever cable you want.
My first guess is that your joists might run in the direction of the red arrows (fireplace on gable end or side of house). If you need to bring the cable in from that side of the house then you'll have to get past or through those joists somehow. This might be where seeing how those vents are ducted comes in handy; if your joists DO run front to back and that IS from the fireplace as a heat exchanger, then they would have had to have provided some way to vent the heat across the joist runs. Now, if your joists run the direction of the green arrows then you could possibly just fish cable through one of those joist pockets all the way to the side wall, and out.
Can't really say much more than that atm without knowing more. If I came out to your house to install cable that would be one of the first things I'd look at (as I hate just running stuff like RG-6 all willy-nilly like cable companies do).
I will say though that most people (especially homeowners) don't worry about plenums at all. NEC Article 300.22 (C) [yes I actually pulled the codebook out
] talks about running cables through 'plenums': this includes cold air returns used in many homes where the return itself is simply one of the joist pockets. By
CODEyou cannot use such a plenum/return air space as a 'raceway' to run your cables - if there IS a joist pocket that is being used as a return air, you have to try to run directly across (perpendicular) through it, or avoid it completely. This is a fire/toxic fumes thing but they know they can't eliminate it 100% from residential so that's why they have the exception. Also by code if you get actual plenum-rated RG-6 you can skip to article 820 or so and toss a bunch of that out, but no homeowner is realistically going to pay for plenum rated RG6. Looking at bulk RG6 prices, 1000ft of standard RG6 might be $50-$60 while 1000ft of plenum rated might be as high as $400-$500. So now that I've said my piece, you can choose to do what 99.999% of all homeowners do and use regular RG6 everywhere, including up through walls in cold air returns to get from, for example, basements to 2nd floors....just be aware that this is a fire hazard though as cable can help spread fires through air returns and it's why the NFPA fights against it being done.