I have to warn you that your estimate on the amount of time to finish Rudin (if done correctly) may be off.
Here's why. Up to now, you've taken the standard advanced course in high school mathematics and done quite well. This is a feat to be proud of, and unfortunately, you've done so well that you are a year ahead of the game. I say unfortunately, because the next natural step would be to take a proof based math class and learn the fundamental skill of writing clear, coherent mathematical proofs. It doesn't matter the subject through which this is done, but this is the step that should happen next.
The problem is this next step is difficult (if not detrimental) to take alone. You need someone to read your proofs, to make sure your arguments make sense and are understandable to another person, and to check that your sentences end in (goddamn) periods.
You can't do the exercises in Rudin (and for that matter learn basic analysis)
without having the skills of proof writing. And for that reason, I advise you to try to find someone to help you acquire this skill. Here are three ideas.
(1) Where are you from? There may be math classes at a local university you can take and get credit for. This will have the added benefit that you will meet other people who like math. Talking about Math is a lot of fun. And while, many mathematicians learn a great deal through self study, it's typically in the context of a mathematically inclined environment. It might be surprising to learn how much of the stuff you think you know is wrong when there is someone there you try to explain it to.
(2) If that fails, try to find a correspondence course. This way you at least get feedback and keep the postal service afloat.
(3) Find a teacher at your school. Many (maybe all) were probably math majors at one point, and could read over your proofs and give feedback.
However, if none of these options are available, I would advise you to stick to the more computationally minded brand of mathematics that you have seen in calculus and differential equations. There are great treatments of linear algebra in this vein. Try Gilbert Strang's 'linear algebra and applications' which has an associated lecture series on MIT open course ware. Another option is to try to learn some programming. Java's great. And tackling a programming problem will stimulate you in a way you might have once thought was reserved only for mathematics.
If all else fails. Fly a kite, learn to surf, and prefect a secret BBQ sauce recipe. It's your last year of high school! Live It Up.
First of all: you shouldn't give up on problems after 30 minutes. Take a break, try a different problem, maybe wait a few days and try again -- you'll gain a lot more from the problem if you struggle and solve it yourself. Having access to solutions can be helpful, but you don't want to find yourself relying on them. (There's a phrase that gets thrown around a lot: "If you can't solve a problem then there's an easier problem you can't solve; find it").
Baby/Blue Rudin is a great book for an introduction to the basics of analysis (beyond one-variable "advanced calculus"). After that I'd suggest looking at the 'Lectures in Analysis' series written by Elias Stein and Rami Shakarchi (Stein was actually Terrence Tao's advisor). These books cover introductory Fourier analysis, complex analysis, measure theory, and functional analysis. Along the way the authors expose you to all kinds of in-depth and enlightening applications (including PDEs, analytic number theory, additive combinatorics, and probability). Of all the analysis textbooks I've looked at, I feel like I've gained the most from the time I've spent with Stein and Shakarchi's series -- these books will expose you to the "bigger picture" that many classical texts ignore (though the "classics" are still worth looking at).
I've skimmed through parts of Terrence Tao's notes on analysis, and these seem like a good option as well (though I looked at his graduate-level notes, I don't know if this is what you're referring to). I've always gotten a lot out of the expository stuff written by Tao, so you probably can't go wrong with the notes regardless. If you feel like you need more exercises, don't be afraid to use multiple books! Carrying around a pile of books can get annoying, but it's always helpful to see how different authors approach the same subject.
Best Answer
Historically speaking, perhaps an analysis course should begin with Fourier's approach to problems regarding the conduction of heat. (For more on this, see e.g. here.)
Of course, many math curricula are organized and re-organized (and re-re-organized) in ways differing radically from their material's original development. For example, Galois Theory is often offered as a sort of "Abstract Algebra II" course, though its creation (e.g. work by Galois himself) did not rely on the group theoretic arguments that you would now likely learn first in an "Abstract Algebra I" class.
As a matter of personal preference, I think it's helpful to start building a topological vocabulary in your first real analysis course (e.g. words such as those mentioned above by Robert Israel). One could argue that learning these definitions in a specific context will be confusing when moving to a more general context (e.g. what compact means in a real analysis course vs. what it will mean in a topology course). Others might argue that having specific examples will help when you go on to study objects of greater generality.
There are a lot of topics that could be covered in an introductory analysis course, and it's quite possible that what is deemed important enough for inclusion is related mostly to the instructor's background (e.g. someone with a background in harmonic analysis might take a more Fourieresque approach in lieu of introducing concepts from point set topology).
I cannot tell you what "should" be done, but I would include a portion on the standard topology in Euclidean space in my own course on analysis, partly because I think the terminology should be seen sooner rather than later, and partly because I find the material particularly interesting and fun to play around with.