My TV is like 10 years old. Resolution means nothing to me

Well I usually go for max resolution first, and then start tweaking the graphical settings.

Minimum resolution (1080p or 900p with good AA) > more demanding graphical features.

Anything above that resolution has marginal benefits for the size monitors I play games on, so then I'd prefer better graphics.

It bothers me if a game has pretty nice graphics, but I can't appreciate them because of image quality artifacts like aliasing, dithering, and screen tearing. These break immersion much more for me than ugly textures, geometry, or pre-baked lighting.

Well, having a low-end PC, I have to make this trade off all the time.  I usually put all my settings to High first and then adjust the resolution accordingly to get the desired frame-rates.

 Besides, Im not sure if it is because I have a relatively small monitor (22") but I can barely notice any difference between 720p and 1080p.  I assume that the bigger the monitor, the more noticeable the difference would be.

And now an example:

Recently I have been playing Portal at 1080P, everything Ultra High and 8x AA.  And Doom at 720p with Medium settings  and no anti-aliasing.

And Doom looks a lot better in my opinion.  So yeah, I prefer graphics over resolution.

Graphics. Though there's a limit. I don't want to be paying at 240p on a tv screen.

I'd much rather better textures/effects/shaders/etc at 720p than worse assets at 1080p.

On Wii U for example, Mario Kart 8 and Captain Toad Treasure Tracker look better at 720p than Smash Bros does at 1080p.

Is better framerate an option? Cause both of those don't matter anymore.

Frame rate, graphics, load times, control remapping > resolution

Resolution and graphics are closely tied together. A game made to look good at a particular resolution will look plain and empty at a much higher resolution. Drop the resolution too low on a game made for a high resolution and you'll miss important details.

It depends on the game: Polybius looks amazing to me, as in much better than Robinson: The Journey on base ps4, resolution + frame rate > graphics. I also prefer to look in the distance, where I'm going in racing games, so again resolution + frame rate > graphics

For RPG / Action Adventure / Horror games graphics > frame rate > resolution. The story is in the world details, so put in as much detail as possible.

Graphics, since it would be nicer on a smaller screen which I am fine with