This is fairly basic introduction to Principles of Analysis, on intermediate undergrad level, strictly in R^1. The only other similar book I'm familiar is Kirkwood. The books of Rudin, Apostol, etc present the subject on much higher level.
My original intention was to take a course with Rudin, but after I've realized I had a hard time digesting his style, I've decided to take more elementary course. I knew the course would be using Lay, so I got this textbook and tried to learn it on my own, but wasn't sure how I was doing and ended up taking the course (still with Lay) anyway. So I'm quite familiar with this textbook. The only topics we didn't cover is "series" and "sequences and series of functions".
Now overall I would say it's a mixed bag. First, the good things. The first few introductory sections on sets and proof techniques are excellent, highly recommended, that's how I learned how to prove. I found exercises very useful.
Now things I don't like. First, lots of typos. I think I had 4th edition, and still I've managed to find over 20 misprints, incorrect references, etc, etc, all were reported directly to author. Second, and that's probably more important, in several instances the proofs are too convoluted and not self-motivating. To be more specific, the proof of Heine-Borell theorem is less than adequate. It is correct, but that's the kind of proof you read and then entirely forget how it went. I remember on the first reading I didn't feel comfortable with this proof at all. When I discussed this book with professor I was going to take that course with, he (surprisingle) agreed with me and told me he would present a different proof (and he did, much better one). Another example: proof that the modified Dirichlet function is Riemann-integrable. The proof can be substantially simplified. In fact, I've managed to simplify it. Finally, the same professor told me Lay's presentation of Riemann integrals had some holes in them, so he used Kirkwood instead. In fact he told me he was making choice between Kirkwood and lay (but ended up choosing Lay because he didn't like Kirkwood's book layout. Kind of funny reason, I think.)
In any case, I think Kirkwood is a bit better for self-study. Unfortunately it doesn't have intro to proofs, logic and sets. Ideally you should have both books, if you plan for self-study.
(note: I did took the Principles of analysis, after I've finished that one with Lay, and did quite well.)