Baby Rudin vs. Abbott

They are both rigorous in that they both give complete proofs of their results. Rudin's problems on the other hand are challenging to newcomers. Abbot's problems are on a much lower level than Rudin's. I love Rudin's books, but there are mixed opinions on whether they should be used as introductions. I used Principles after my first year of analysis and loved it. I'd say first work through Abbot because he will likely provide more motivation. Later, get Rudin and push your boundaries of understanding. You might just become an analyst after that approach. It's what happened to me.

You think you are asking just a simple question but you are adding kerosene to the flame of Pro-Rudins vs Anti-Rudins :) Any ways I have read both books and here are my feelings

(1) Rudin definitely has really good, challenging problems

(2) Rudin goes directly to the point

(3) Rudin gives you the opportunity to think on your feet and fill the gaps

That said coming from a normal math background, Rudin was a tough read for me. Abbott's book was quite a good supplement for some chapters. But for me Pugh's Real Analysis and Apostol Analysis was the best supplement. My strategy was to read and re-read Rudin and give some critical time $t_{c}$ to see if I understand a concept. If that doesn't happen $t>t_{c}$, I refered to the other books. Towards the end of the course, I came to know Terry Tao's book and I have to say it was a pleasure to read. Good luck, you can always pick one book and refer to the others when you need to. You need to have a general idea of where you are going but you don't have to follow every route

Can't do better than read "Bolzano Bourbaki's" review of Baby Rudin on Amazon.