You've gotten some good advice so far; permit me to chime in with the perspective of someone on the other side of the lectern. I've been teaching at small liberal arts colleges for longer (far longer) than any of my current students have been alive and over the years my colleagues and I have seen many of our majors accepted at good-quality graduate schools, both for Master's and Ph.D. programs. Let me get your major question out of the way immediately
[I]f you go to a small school, are you doomed to fail right from the get-go?
This is easy to answer: no! as I've mentioned above.
Now that that's out of the way, I first need to back up a little and ask you the question you should ask yourself,
Why do I want to get a Ph.D.?
The process of obtaining a Doctorate is long and, frankly, painful. If you've decided put your life on hold for five to seven hard and exasperating years, you'd better have a good reason. If your goal is to continue doing mathematics and eventually producing results of your own that add to the body of what is known, that's commendable. If you view the Ph.D. as a high-falutin' teaching certificate and see yourself as primarily a college professor, that's also commendable. If, on the other hand, you're contemplating merely drifting into a way to keep doing what you're good at, it might be a good idea to ask yourself whether there might be some other way to spend the next few years, since a Doctorate in mathematics won't help you very much in the Real World outside of academia and may actually harm more than help in eventually landing a job out there.
For the purpose of discussion, let's assume that you have thought long and hard and decided, "Yup, I really want to get that diploma," how can you get your foot in the door? You've earned superb grades in what looks to me to be a reasonably solid undergraduate curriculum. For all but the applicants for the very top schools, that will work in your favor, though you're correct in your assessment that good grades will be a small part of the admissions committee's view of you. Similarly, the GRE will also be a small part of your total package---those scores will primarily be a check for the Committee to ensure that you're not a hopeless idiot. On the other hand, you probably won't have any Fields medalists among your letters of reference, but that's not as bad as it may seem, especially if any recent graduates from your college have made it into a school for which you're applying.
Reference letters are the single most important part of your package, hands down. No matter where you did your undergraduate work and no matter who's writing your letters, the sentence "X was the best student I've seen in ten years" will loom large in the eyes of the Committee. If your English is excellent, make sure one or more of your letters mentions that, since universities are always on the lookout for good teaching assistant material.
Is there an area of math that really excites you? If so, check for universities with strong programs in that area and mention that in your personal statement. It's already been mentioned that it might be a good idea to write to faculty members at your target school and ask about what they're doing in the area of your interest before you submit your application.
That said, where should you apply? I agree that a top-tier graduate school is probably a stretch, though that shouldn't rule out trying for one or two. Who knows, you might get lucky? On the other hand, you probably shouldn't limit yourself to universities whose motto on their seal is "Plenty of Free Parking." There are plenty of respectable universities between those two extremes and for the time being they still need a crop of good applicants (which you are). As I said, we've sent a lot of students off to grad schools and I can't recall a case where one of our students failed to get into at least one of the schools to which they applied. Your task over the next few months is to research those middle-tier schools that look like the best fit for you.
As Ragib and Francis mentioned, getting a Master's degree first isn't all that bad an idea. At least, it might give you an idea about whether going on for the Doctorate is what you want to do (that's what I did, though for different reasons). If you go that route, do your best to make yourself stand out from the crowd, so your subsequent letters will reflect that. Keep in mind, though, that you'll have to dig up the money to pay for your education, since graduate schools rarely offer financial support for Master's students (though many will provide full support for Ph.D. students).
Finally, I fully agree with some of the other answers: don't be discouraged. Things are nowhere near as bad as you've painted them. If that's what you want, you'll almost certainly succeed. Best of luck---keep us posted.
Best Answer
Just my thoughts:
I think SDEs and PDEs are rather deeply intertwined. For instance, the Kolmogorov backward equation and Fokker-Planck equations (see the link in the comments above). Indeed, as expected intuitively, Ito diffusion processes and the classical diffusion (heat) PDE are deeply related: on a Riemannian Manifold $(M,g)$, if $X$ is Brownian motion on $M$ (i.e. an Ito diffusion process with infinitesimal generator $\Delta_g/2$) with some transition density function $p(x,y,t)$, then $p$ solves the following: $$ \frac{\partial p}{\partial t} = \frac{1}{2}\Delta_g\, p,\,\;\;\; \lim_{t\rightarrow 0} p(t,x,y) = \delta_x(y) $$ which is of course a heat equation (see e.g. Hsu, Heat Equations on Riemannian Manifolds and Bismut's formula). In the simple case of Euclidean space, we get $\Delta=\Delta_g$ and $$ p(t,x,y) = \frac{1}{ (2\pi t)^{n/2} } \exp\left( \frac{-||x-y||^2}{2t} \right) $$ which is the Gaussian heat kernel of the IVP above (see e.g.: Morters & Peres, Brownian Motion).
Well, beyond PDE/ODE theory itself, which is much more commonly applied (at least for biological models), you get the outlooks from (1) above. :)
As for what to study, it depends on your goal, but I would look at dynamical systems.
Dynamical systems, ODEs, and PDEs are important tools for ecological, physiological and cellular (since they are often physically based, more in the biomedical engineering realm), and molecular systems modelling (see "systems biology"). But one area I know with heavy attention paid to stochastic partial differential equations (SPDEs) in particular is in modelling neurons (e.g. see Tucker, Stochastic partial differential equations in Neurobiology: linear and nonlinear models for spiking neurons) because the ion channels are noisy and membrane voltage is a function of both space and time.