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.
While I am not familiar with the book by Schaum, I think you are on to something. There is definitely a difference between textbooks aimed at high school students and textbooks aimed at college students. Furthermore, this isn't just true of mathematics, but of other subjects as well. I would go as far as to say the whole learning process is generally a little different at the college level than at the high school level.
Part of this difference is made up by a change in how the information that you are expected to know at the end of a course is presented.
In high school new information is presented to the student verbally, through the medium of one's teacher. Ideally the teacher also serves as a motivational aid, but the textbook is relegated to a marginal role as a supplementary aid and reference. While textbook exercises are very important, they usually do not present new concepts, but merely help a student internalise already familiar ones. Perhaps high school textbooks are generally light on theory because it is assumed that the teacher will present new concepts during class discussion, and that many students will not bother to read the text.
In college new information is transmitted mostly via the written form, through the medium of one's textbooks (and, especially later on, papers and journals). The professor thus becomes relegated to the marginal role of a supplementary aid and motivator. This becomes increasingly the case as one progresses to the higher levels of college education. In fact, some, like the eminent philosopher David Hume, completely dismissed the usefulness of professors as knowledge repositories: "There is nothing to be learned from a Professor, which is not to be met with in Books." I would not go as far myself, but very often the professor's most important role is to put the right books in one's hand, and to motivate one to read them, as your mathematics teacher seems to have done. Plus, someone has to write the books.
What is certain is that you will be doing a lot more reading at the college level. This is true for the subject of mathematics and for other subjects as well.
A few years ago I came across a book that can help with this transition in acquiring information. It is called How to Read A Book, by Mortimer J. Adler and Charles Van Doren. Adler was a writer and philosopher who was the main force behind Encylopedia Britannica's 60-volume Great Books of the Western World series, many of which are mathematics textbooks, or at least mathematically themed (e.g. Euclid's Elements, Apollonius of Perga's On Conic Sections, Nicomachus' Introduction to Arithmetic, Descartes' Geometry, Newton's Principia Mathematica, Whitehead's Introduction to Mathematics), which suggests that he wasn't unfamiliar with the subject. The book has been in print since 1960 and the latest edition is from 1972, but its subject matter is timeless. Books have been around for a long time, and will no doubt continue to exist in some shape or form in the foreseeable future.
At first it might seem a little silly to read a book about reading, but the subject is approached quite formally, and the reader is given a series of rules and guidelines that are to be observed when reading analytically, or reading certain types of subject matter, such as mathematics. Many of these rules were completely new and extremely helpful to me, despite considering myself an avid reader prior to tackling this book. If you read this book carefully, and one can do this especially well if one recursively applies the concepts presented within the book to the book itself, you will emerge a much more analytical reader. This will greatly help you handle both mathematical text books at the college level, as well as books dealing with other subjects.
How to Read a Book deals with all kinds of reading matter, but is especially useful in handling expository books, or, to use your terms, books that contain large amounts of "theory". There is a specific section on how to read mathematical books. The critical reading section, however, which makes up the majority of the book, applies to all expository books, of which mathematical books are merely a subset.
I wish you great success on your quest of mastering Schaum's book, as well as any others that may come after.
Best Answer
As others have pointed out, Rudin's book is a little bit hard even for people who has more mathematical maturity. You should try some other alternatives, for example:
It's perhaps a good idea to use:
This book explained me a lot about the hierarchy of the proofs in Analysis and what were the challenges met by the people who created it. Another interesting read is:
Also, take a look at some of the recommendations in the MAA Reviews. One interesting review is the one on Rudin's book. I'd follow Arnold's advice, the book he recommends is superb.
Now beyond these historical perspectives on analysis, you might find this book useful:
You might also like the following book:
It's not too related to analysis in the sense of Rudin's book, but I think it's illuminating for the subject.