The Real Poop
When you are a kid and you hear that someone's dad is an engineer, you're thinking about The Little Engine That Could and the guy with the funky hat and suspenders who runs the steam train up up up the hill for all the good little boys and girls in WhateverTown. Different kind of engineer. This one couldn't give a rat's sass about little boys and girls. (We are sure some aerospace engineers are actually quite fond of children. We just mean to say that it isn't in their job description.)
Engineers "figure stuff out" (technical definition). They solve problems. Lawyers engineer the legal system. Doctors engineer your biology. And aerospace engineers figure out ways to make stuff work in air and space. Take that, doctors and lawyers.
So the question you have to ask yourself is, "Do I feel spacey?" "Do I have what it takes to be a rocket scientist?" "Do I get excited each day figuring out how to get satellites to spy on people?" Or some iteration of the preceding.
More Low-Down: More Poop
Aerospace engineers are employees, usually of the government (like NASA if it's still around in 10 years—this is where you can really make mondo space bucks), and also of semi-private industry like Boeing and Rockwell and the gang. On the low end, you could be working for a Products and Parts Manufacturing company, but you still wouldn't be working for nuts and bolts.
The business of space flight is so expensive and so complex that engineers usually have to specialize in incredibly narrow areas. Yes, even narrower than that crawlspace underneath your basement stairs. We're talking so-narrow-you-couldn't-slip-a-piece-of-paper-in-there narrow, and with that kind of specialization comes career risk.
Say you specialize in solid fuel rocket booster connector ties. It's a great gig, until one day, when the world shifts to liquid fuel and the one narrow thing that you spent twenty years "owning" is no longer relevant. The company you work for shutters your division. Nobody else wants to hire you because you only know about outdated tech. What's your next job if not asking, "Yessir, La Guardia or JFK?"
Provided that your niche doesn't suddenly up and vanish, though, you're in better shape than most of your engineer brethren. You'll be making more money than materials engineers, petroleum drilling engineers, chemical engineers, and nuclear engineers. No, not more than all of them combined. Let's not get greedy.
Most aerospace engineers are really smart—although not always business-savvy—people. Many have sacrificed money and freedom to do something amazingly cool. Spending an entire day calculating to the ounce how much fuel will be needed to lift a lunar rover off the surface and get it just far enough to be captured by the Earth's gravity can be a reward in itself. And if that's not enough for you, we can have a cheap plastic trophy made for you that reads: World's Best Aerospace Engineer. You can keep it on the edge of your desk, right next to your I'd Rather Be Studying Thermodynamics coffee mug.
If you happen to be one of these smart guys or gals who really would rather be studying thermodynamics, loves airplanes and/or space shuttles, and is eager to understand all the minutiae of how complex aircraft work (most of us just board some big metal machine and blindly trust that it will get us to where we need to go), you could be a born aerospace engineer. If you're only interested in space because you think Jupiter looks pretty, you might prefer being an astronaut. Or a painter. Hey, it takes all kinds to appreciate the cosmos.