So, now you're at university, and you're headed for grad school ...
(the following is horribly UScentric, 'cause that's where I am right now, the general principles are broadly applicable, the actual getting into grad school procedure bit in future post will be both US and THEM centric), now what?
Well, each cohort in the US is about 4+ million people, about 4000 of those major in physics. Since participation in the further education in the US is almost 50%, that is 4000 out of about 2 million, or 0.2% of undergraduates (specifically, about 1.2 million bachelors degrees are awarded each year, with physics major 0.34% of those, near historic lows, trend has flattened after many years of decline, see APS jobs in physics stats. For what it is worth, about 1200-1500 PhDs in physics are currently awarded each year, almost half to non-US students, so about 1/6 to 1/7 of undergrads end up doing a PhD. The "rule of thirds" you'd infer from the raw numbers (# PhDs = 1/3 # BSc) is surprisingly robust at each step.
The number of astronomy undergraduates is much smaller still, since not a lot of universities (61 according to this article by Cabanela and Partridge in AER) have a separate astronomy department or a separate major (so a lot of people who become astronomers or astrophysicists start as physics majors, or have an astronomy minor). The number of astronomy PhDs is fluctuating around 120 each year, with significant Poisson noise, as one might expect. There are 25-30 major research universities which dominate the astronomy PhD production, so classes are small, and the intake is fought over hard.
Having said that, current astronomy undergraduate production is about 300 per year, and has grown significantly AIP stats summarises. A lot of those are double majors with physics. Naively them about 1/2 of undergrads go on to PhDs, but after you allow for the foreign intake, it is more like 1/4 - 1/5, since some also go into graduate school in physics (and of course other fields, but I am not considering those now).
So, the good news is that if you're in the major, you have a high prior probability of going to grad school. You just have to pass the classes, get good grades, not go broke, survive the insanity of university, enjoy life, and take the GRE exam, the only exam stupider than the SATs...
So, what should you do. First, take all the math classes you can handle, especially if your interest is astrophysics; take calculus of course, through ODE, PDE, Complex Analysis, and some course covering spectral methods (Fourier and Laplace transforms etc). Better know what a Bessel function is when you get to grad school, and know immediately where to start on solutions to second order differential equations. Take also probability theory and statistics, linear algebra, numerical analysis, and computational techniques. If you can stand it, differential geometry, topology, functional analysis and some advanced classes never hurt anyone.
I know I keep hammering on this point, but the primary limit on career paths within astrophysics is inadequate math preparation - too many students take minimal load of math classes and skip mathematics that is "not relevant" and then find years later that they did need if after all but they don't have the time, inclination or energy to go back and learn on their own.
Then take physics. You don't have to take all of it, but would it hurt to do so? Solid state physics, wave theory, optics and quantum field theory all show up somewhere in astrophysics (all of physics does, that is why astrophysics is the greatest sub-field in physics! ;-) oh, and take relativity if you're at one of the few places it is offered.
What else. Well, the prereqs and college mandated hoops of course. Don't burn yourself out, but given a choice, wouldn't you rather take a gen ed class on something that is interesting (to you). Funky minors sound fun and macho, but they can destroy a GPA, a graduation schedule and enthusiasm - approach with care. Then think about what you like to do when you finally get there? Like to build things, add labs and engineering classes (especially electronic and optics). Like data, add statistics and computing classes. Like to think, go buy a ruled pad and nice black ink pen (no pencils, this isn't kindergarten).
So, you did that, aced all the classes, are student president, captain of some NCAA team, and still have time to party.
Well, a long time ago, there was a rule of thumb - the odds of an incoming undergraduate in physics/astronomy getting a faculty position at a research university is about the same as the odds of an NCAA division 1 football player making the NFL. But the earning potential is different (not as much as you'd think, lifetime average!). And in both cases, it is not a bad career move, even if you don't make it all the way, the alternate career paths can be sweet. Oh, and in both cases, where you go to school matters a lot, for your average odds. But remember, there is always a Jerry Rice!
So, end of your junior year, if you can still stand the field, you need to start seriously working on grad school applications, and planning for the GREs. More later...
There's some very good advice in the post above (I'm a physics graduate student), but remember not to get discouraged if you havn't done all of those things. Everyone's experiences are a little different, and the system has a way of accomdating most people who want it. It's easy for an established professor to talk about exactly what the model undergrad would take, but I know when I read stuff like this, as useful as it is, it's easy to say, "crap, I only met Bessel functions for the first time this year and I'm in grad school, I must be some kind of freak".
I should add to that: if you went to a liberal arts college and didn't have access to many of the courses mentioned above, don't throw in the towel. The advantages of such an education are hard to quantify but make you a more rounded human being and stick with you. You'll be at a disadvantage, but you can still go to grad school.
Seriously, the reason I mention stuff like that is precisely because so many students end up with less than ideal (from my perspective) backgrounds.
Now, I care about Bessel functions, but there is only so much time, and people also need lab experiences and gen ed and so on and so forth.
My concern about math is that in my experience it is the subject people have the worst preparation in coming out of high school in the US, which means physical scientists who are not math majors are always behind the curve on what they are expected to know; and, it is the subject most people have the hardest time picking up later.
Well, at least it's harder for a grad student to blow out his knee, thus ending his career. I guess there's always mad cow disease for us book learnin' types.
It's not that this isn't good advice. Everyone that is interested should heed it. But lots of people don't work out what they want to do until after they've missed the "ideal" undergraduate preparation and I hate to see people get discouraged - I know I often found perfect-in-hindsight advice like this frustrating when it was too late for me. I'm only trying to point out there's still hope.