Michael Spivak's Calculus
This is really an honors level calculus text, but it might be useful to have around. It's pretty expensive, however.
Stephen Abbott's Understanding Analysis
Back in January 2003, in sci.math, I wrote: I think it's the best written introductory analysis book that's appeared in the past couple of decades.
David M. Bressoud's A Radical Approach to Real Analysis
Takes a more historical approach than any other standard text currently in use. Heavy focus on pointwise and uniform convergence of sequences of functions (power series, Fourier series).
Victor Bryant's Yet Another Introduction to Analysis
All exercises have solutions. Full of pictures, diagrams, and other pedagogical aids. Excellent as a direct sequel to the standard calculus sequence and the book is carefully written with this in mind.
I've been thinking about your specific concerns and I cannot think of anything inherently unique about your situation or your background that would suggest a specific approach. My feeling is that typical differences among students in general would be first order effects, while issues facing someone coming from a computer science background would be second order effects that get drowned out (at the population level) by the first order effects. In fact, your background in number theory and combinatorics probably make you better prepared for real analysis than many (most?) students are after the standard calculus, ODE, and linear algebra courses taken in the first two years of college.
What I suggest is that you visit a university library, bringing a pen and paper to write on, and spend two or three hours browsing the shelves. Real analysis texts will be mostly located in the vicinity of QA 300 to QA 330 Library of Congress classification. Read the prefaces of the texts on the shelves, and perhaps even photocopy the prefaces along with the tables of contents. Jot down titles of books that look interesting, either for primary study or for secondary reading. Make a note of how frequently the book was checked out or how worn out from reader use the book seems. Then do some online searching to see what others say about the various books. Keep in mind that a book no one talks about could still be very good. Most people are only aware of the books they used and perhaps a few other books, and most people commenting online probably have little knowledge of anything that hasn't been widely used in the last 30 years. For example, I think John F. Randolph's Basic Real And Abstract Analysis is one of the best beginning graduate level real analysis texts I know of, but because it appeared in 1968 and because it is an awkward fit for most mathematics programs (too advanced for most undergraduate courses, too leisurely for most graduate courses; in this respect it is a bit like Neal L. Carothers' text Real Analysis), there is virtually no discussion of Randolph's book anywhere online.
A good time to visit such a library is a week or two after final exams, because a lot of books that had been checked out will be returned and the library staff will have had time to reshelve the books.
Early this morning I went through a lot of papers I have related to the teaching of real analysis and made a list of those that could be of use to you if you're especially interested in discussions about students' difficulties in real analysis and people's thoughts on how to minimize these difficulties. I am not including papers that would be mainly of interest to those whose primary focus is on mathematics education rather than on mathematics, because despite this being a mathematics education group, your interest is in mathematics. [The difference for me is whether the paper appeared to be primarily about mathematical ideas or to be primarily about psychological and sociological aspects of learning mathematical ideas.]
ADDITIONAL BOOKS THAT MAY BE WORTH LOOKING AT
 Robert Lee Brabenec, Resources for the Study of Real Analysis, Classroom Resource Materials, Mathematical Association of America, 2004, xii + 231 pages.
 Andrew Michael Bruckner, Judith Brostoff Bruckner, and Brian Sheriff Thomson, Elementary Real Analysis, Prentice-Hall, 2001, xvi + 677 + 58 pages.
The final 58 pages consist of appendixes: (A) Background (B) Hints for Selected Exercises (C) Subject Index. Like Bressoud's book above, this book does not suffer from the problem pointed out by Bressoud in his review of another book below. Also, it's now freely available on the internet.
 Kenneth Ownsworth May (editor), Lectures on Calculus, Holden-Day, 1967, vii + 180 pages.
Contains some very interesting selections, written by 9 different people. I encourage looking at this if/when you visit a university library to browse real analysis books. Some (but not all) of the chapters will probably be too advanced for a beginning real analysis student.
 Joanne Erdman Snow and Kirk Edward Weller, Exploratory Examples for Real Analysis, Classroom Resource Materials, Mathematical Association of America, 2003, xv + 141 pages.
 Robert Michael Young, Excursions in Calculus. An Interplay of the Continuous and the Discrete, Dolciani Mathematical Expositions #13, Mathematical Association of America, 1992, xiv + 417 pages.
This book is an extremely well-cited encyclopedic exposition with 463 references, one of which is Young's book itself. In addition, on pp. 397--408 a reference citation is given for each of the book's 417 problems (or the problem is stated as currently unsolved).
MISCELLANEOUS ITEMS RELATED TO THE TEACHING OF REAL ANALYSIS
 Teaching Mathematical Analysis to Students, Teaching and Learning Undergraduate Mathematics (TALUM) archive.
Articles written by members of the TALUM content working group: Johnston Anderson, Keith Austin, Janet Jagger, Frank Jellet, David Tall.
 David Marius Bressoud, Review of Undergraduate Analysis: Second Edition by Serge A. Lang, Mathematical Intelligencer 20 #1 (Winter 1998), 76-77.
From Bressoud's review "Throughout this book, all of the motivations and the intuitions and the connections have been wrung out of the mathematics." The 2nd half of Bressoud's review discusses his experience with teaching a second semester undergraduate course in real analysis that made heavy (but not exclusive) use of Thomas Hawkins's book Lebesgue's Theory of Integration. Its Origins and Development.
 Thomas Arthur Alan Broadbent and others, The first encounter with a limit, Mathematical Gazette 19 #233 (May 1935), 109-123.
This is a written report of an extended discussion on introducing limits to students that occurred at the January 1935 Annual Meeting of the [British] Mathematical Association. Most of the participants were teachers at various university preparatory schools, the exceptions being Broadbent (at University of Reading) and brief appearances by Godfrey Harold Hardy and George Neville Watson. See also the letters to the editor in the same issue of this journal by F. C. Boon (on pp. 131-132) and N. M. Gibbins (on pp. 132-134), as well as Dockeray (1935) below.
 N. R. C. Dockeray, The teaching of mathematical analysis in schools, Mathematical Gazette 19 #236 (December 1935), 321-336.
This is a written version of a paper given on 23 February 1935 at a meeting of the London Branch of the [British] Mathematical Association. Immediately following this article, on pp. 336-340, is a written report of the discussion at the meeting that followed the talk. Related to this is Broadbent (1935) above.
 Marianne Freundlich, Eventually, Notices of the American Mathematical Society 45 #5 (May 1998), 599-600.
Personal reflections about the author's struggles to understand epsilon-delta definitions and proofs in analysis.
 Gurcharan Singh Gill, A survey of texts for a one year course in real analysis at the advanced undergraduate or beginning graduate level, American Mathematical Monthly 75 #9 (November 1968), 1033-1035.
 E. J. Moulton, The content of a second course in calculus, American Mathematical Monthly 25 #10 (December 1918), 429-434.
Dated, but contains useful observations about the teaching of certain topics in real analysis. Note the follow-up response: David Clinton Gillespie, Advanced calculus or differential equations, American Mathematical Monthly 26 #5 (May 1919), 189-190.
 Edgar G. Phillips, On the teaching of analysis, Mathematical Gazette 14 #214 (December 1929), 571-573.
 Kenneth Allen Ross, Genesis of elementary analysis courses, Notices of the American Mathematical Society 60 #9 (October 2013), 1179-1180.
 Henry Scheffé, At what level of rigor should advanced calculus for undergraduates be taught?, American Mathematical Monthly 47 #9 (November 1940), 635-640.
 David Bernard Scott, On teaching analysis, Mathematics Teaching 42 (Spring 1968), 34-41.
 Rolph Ludwig Edward Schwarzenberger and David Orme Tall, Conflicts in the learning of real numbers and limits, Mathematics Teaching 82 (March 1978), 44-49.
 David Bernard Scott, On teaching analysis, Mathematics Teaching 42 (Spring 1968), 34-41.