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
The most important issue I see here is to get yourself ready for a PhD program in probability. Depending on what is available in your current program, presumably an undergraduate major in mathematics, you should consider the following. Make sure you really know calculus through multiple integration and and infinite series. Take a course in real analysis. Learn something about measure theory. Take a mathematical statistics course (or at least a statistics course with a calculus prerequisite). Find out what Bayesian statistics is. And a course in linear algebra would certainly be useful.
Look at the web sites of departments to which you might feasibly be admitted; some of them give specifics of the background they want applicants to have.
You will be exposed to a lot of new ideas and possibilities in a PhD program in probability. The CS related fields you mention are changing so rapidly and progress in them is tightly held and not always public information, that it might not be possible for you to prepare yourself for those fields now. It is entirely possible that your objectives will change as you progress towards your PhD in probability. So my suggestions along lines of the CS topics you mention are more tentative and possibly not as important as my suggestions on preparing for the probability PhD.
Of course, you should take obviously relevant courses in computer science. Some less obvious possibilities are number theory, group theory, and quantum mechanics. Learn to use a statistical programming package/language such as R or Python (with Scripy)--data management, standard statistical analysis, simulation (e.g., permutation tests, bootstrapping, stochastic processes, maybe even Gibbs sampling).
Outside of coursework, try to keep current in developments in various kinds of artificial intelligence, and proposals for making the Internet more secure. You may find out more about these things by reading the New York Times than from CS courses. (Generally speaking, it is a real struggle for colleges and universities to keep current with developing trends in CS because the non-academic job market for potential faculty members is so active.) Much of the innovation in these areas is going on at places like Google and Facebook in the private sector and NSA in the government sector, and you may only be able to get miscellaneous rumors, hints, and clues about the most important developments.
Personal note: All of that said, I went off to a PhD program in probability and statistics out a first rate undergrad program, but with only a one-quarter course out of Feller's probability book to give me a clue what I was getting into. I did survive, and over 50 years later I'm still learning stuff. So I can't say planning is everything. Being at the right place at the right time is important and unpredictable. (But never having the curiosity to be anywhere interesting any of the time is clearly not an optimal strategy.)