I don't have experience with answering questions here, but I have experience with learning mathematics by myself. You'll have to decide if you want an engineering-oriented book or a "pure-mathematics" rigorous approach to calculus.
If the former, engineers sometimes like the heavy glossy-paged books with photos of spaceships in them (like Stewart etc.). They tend to be calculation-oriented, somewhat lax on rigor (proofs omitted or glossed over) with occasional real world applications, and (to their credit perhaps) many graphs of functions plotted out. Albeit almost as many useless photos and flashy design elements. They also tend to be damn expensive.
If, however, you wanna get serious about it, you should make sure you have what Americans call "precalculus" in place. There are many books for this, for example Axler's, which is good but way too long for my taste. You, as a reader, can abridge it yourself - usually it helps not to read math linearly. Also, today you can even learn precalculus on Khan academy. And if you are past that, you might want a sort of general introduction to math, in order to get used to proofs, for example Liebeck's valuable book. Mathematics undergrads receive this intro-to-math material surreptitiously by taking a freshman course in discrete mathematics or elementary set-theory.
Next, there are many options to start learning calculus (ahem, analysis). There are classics that everyone sees recommendations for; I won't reiterate the names of famous apostles and babies, because they are good books but less user friendly than the modern ones.
I've had more success with the (usually non-American) way of approaching analysis by combining, from the outset, what Americans call "calculus" (more calculation oriented courses where you learn to integrate or differentiate various elementary functions, with a pinch of generality here and there) with the material of so-called "Analysis" courses. That is, to learn analysis rigorously in tandem with a healthy dose of specific examples (specific functions, say) and applications.
Hence my first recommendation is weird, and not often heard (it's also not old enough to be a classic): this odd-ball by Canuto-Tobacco, and its sequel. I say "oddball" because the translation from Italian to English is so bad, it's comical. But the treatment is rigorous, user friendly, and with many examples and solved exercises. Some canonical proofs were oddly left out, and are available as "internet supplements" from the authors' website (also in horrible translation) - a heroic attempt to save some paper perhaps.
After that you can go on to more advanced analysis books for which there are many recommendations on this website. I added this "strange" recommendation of a title because I felt nobody else would make it here, and this book has been valuable to me in studying on my own. Some advanced math students may find it too slow, and some engineer oriented students may find it too rigorous - so it's not for everyone. I would say it's for people who are interested in applications (within mathematics) and also in a precise treatment of canonical theories.
Good Luck.
