Personally I would recommend mathematical analysis by Pugh. Goes into as much depth as Rudin, but more approachable and a great selection of exercises.

I like Pugh too. All three are sitting on my bookshelf :). Coming from very little formal math training, and trying to self-study analysis, I would bounce from Rudin if I got stuck, go to Abbott for a confidence boost, and try out some Pugh for a different viewpoint. Pugh might have been just right in terms of rigor and accessibility.

I was under the impression that Pugh, while more approachable, was also much more difficult. Especially with regards to the exercises.