There are countries where people are puzzled if you tell them there is a distinction between "calculus" and "analysis." They think "calculus" is just an old-fashioned name for analysis. The reason these subjects are viewed as different in North America is because a typical "calculus" class is where one learns the mechanical aspects and basic applications of calculus, whereas "analysis" is where you learn everything that comes after that, including relearning the theoretical parts of calculus, but without an excessive focus on the familiar practical aspects.
So to get to my point, most textbooks published in the U.S. or Canada with a title like "mathematical analysis" are made precisely for people like you. For example, you could use Apostol's Mathematical Analysis. Mathematical Analysis I, II by Zorich, translated from Russian, is also very good, and can be used in conjunction with the problem book by Makarov and Goluzina, which has mostly non-routine problems with hints and answers. I think you would find either of these books preferable to Spivak's Calculus, which was originally intended for people learning the material for the first time (though it's not always used that way). One caveat about Apostol's book is that the chapter on Riemann-Stieltjes integration can be a bit difficult for those who haven't studied the plain Riemann integral rigorously elsewhere.
In many North American universities, there is a second course in linear algebra that revisits the subject from a more theoretical perspective. And of course, there are textbooks to match this approach, usually beginning with vector spaces (e.g., Lang or Friedberg/Insel/Spence). However, in your case an attractive alternative might be to study abstract algebra and linear algebra concurrently (vector spaces being a special case of groups, and groups being widely applied in linear algebra in other ways). An excellent book that takes this combined approach is Algebra by Artin. Godement's Algebra is also outstanding, though a good deal drier.
I would consider basic mathematical analysis and algebra at the level of those books to be the foundation of an undergraduate education in mathematics. After that, there are many directions you can go in. I think you'll find that books in differential equations, complex analysis, etc., that are aimed at people who have already reached that level of sophistication are sufficiently different from what you've done before that you won't feel frustrated by them. You can get an idea of what topics are most important by looking at the undergraduate curricula of good universities.
There are also books on analysis and algebra at a lower level of difficulty and covering much less material. Elementary Analysis by Ross and Abstract Algebra by Herstein come to mind. (The latter doesn't discuss linear algebra.)