I am planning to self-study analysis (understood in the European sense, which I think includes calculus).
I have boiled down my book search to three intro books before I move on to further study. To be honest, I am tempted to read all of them:
Spivak's Calculus
Ross' Elementary Calculus
Abbott's Understanding Analysis.
But in which order? Spivak, despite its title, covers pretty much the same ground, albeit in in 500 as opposed to the 300 pages that the other two need. All of them have many exercises and solutions available, which is why I have selected them. What is important to me, that I learn to write good proofs that also give a glimpse of the reasoning behind. The calculus I learned was rather poor and I have forgotten most of it, so I really would like to learn to prove the basics such as limits, convergence, continuity and so on.
Which one to begin with? As time is limited, would reading one or two suffice? If so, which one?
All the best!
As far as I can tell, you've covered the basics of calculus insofar as you've done the exercises of 'determine the limit' or 'calculate the integral' and it all seems familiar. So my assumption is that you want to understand why this makes sense, in a rigorous sense.
Abbott is great for pedagogical reasons, and for your intentions it is perfect. I am inclined to say that you do not need the other two, as the attitude in Calculus and Analysis are different; You want rigour, not necessarily the results. Abbott will cover the results in calculus but in the rigorous approach of how the mathematician understands limits; For example, Abbott emphasizes the attitude that integrals are limits of sums rather than antiderivatives. Calculus will not make you understand proofs, but Analysis will.
For additional depth I'll recommend Rudin's Principles of Mathematical Analysis, as it is written in a 'very mathematical way'. I'll emphasize though that understanding analysis is difficult without having done the calculus.