Spivak or Apostol? Which one of those is the best for a person interested in pure mathematics and who wants to have a deep understanding of calculus? Apostol or Spivak? Could you guys tell me some differences between the approaches of them? What about the exercises? I would like to be challenged, but in a constructive way.
 A: While both books have complete proofs and a good emphasis on theory, Spivak's book is better as an introduction to rigorous math because many of its problems are more difficult and theoretically oriented than Apostol's. (I assume you mean his Calculus). Spivak's book also has a solution manual, which is very useful when you're studying on your own.
On the other hand, Apostol actually covers more material, even just within Volume 1. Volume 2 of Apostol is actually one of the best introductions to multivariable calculus. Apostol's book also has a greater variety of exercises involving applications of calculus to physics.
If you have time, learning from both books is a good idea. Otherwise if your main focus is pure math, then I would recommend Spivak.
A: I don't have experience with Spivak, but Apostol has provided me with a great supplement to the lecture notes of my teacher. The exercises can however be very challenging. 
My personal recommendation would be to start with Stephen Abbott's Understanding Analysis and once you've laid a foundation to move over to Apostol. 
A: I started with abbott Understanding Analysis then move on to Apostol. Stewart don't provide enough rigour but a good starting point for very early beginners into calculus. In my order
(1) Stewart calculus
(2) abbott Understanding Analysis (pretty small text)
(3) Apostol calculus vol 1
(4) Spivak calculus (for more understanding)
(5) Apostol vol 2
