I think mechanical is the way to go to start. You're asking for opinions, so you're sure getting them, but I think that's the most sound.
I think your use of the word "dated" may be throwing some people off, where they're thinking old-school style that looks like it belongs in a different era, but you mean the car looks worn out/weathered. Did I get that right?
Anyway, you can spend to the moon making the car pretty, but since you want it to be able to go somewhere in a hurry and stop even faster so you can put your flaming hair out, I think it makes sense to get all of the mechanical stuff situated before you start turning it into an aesthetic pleasure (aside from sweet engine/exhaust sound, which comes with the mechanical work and makes a good argument for me ripping out the stereo and telling passengers to shut up
).
I know what you mean about the college folks who seem to only focus on looks, or looks before performance. I have to deal with plenty of them myself, but they don't seem to think I'm off my rocker for wanting to focus on making sure my car is mechanically sound before I even bother touching the body. It's not a brand new car that needs nothing. It's a 15-year-old car that needs plenty, but still drives well. You only see the imperfections when you get close enough. From a distance and passing by on the road it looks great, and I'm always more concerned with how it handles and drives than how it looks...though I'll admit to a bit of mental anguish when it's extremely filthy. But hey, at the most I get to see the tops of the fenders through the windshield, and the very top front edges of the rear quarters when the top is down. It's not as important to me as what I'm touching, hearing, and feeling while I drive.