Introduction to Real Analysis books

Rudin's Principles of Mathematical Analysis is a hard book, but it's also a standard and it is extremely well-written (in my and many others' opinion) so you should read it early on. It may not be your first book in analysis, but if not I would make it your second.

When working through Rudin, even though you have a solutions manual, you should not give up on problems before you have solved them. There are problems in that book that take some of the best students hours over days to solve. The process of banging your head against the wall (or the book, or any other hard object) is part of the book and part of your preparation for mathematics. When you do get through Rudin, you will be in a very good place to step into the field of analysis $-$ possibly even the next Rudin book, Real and Complex Analysis.

As a soft introduction to analysis before Rudin, I would recommend my teacher's book: Mikusinski's An Introduction to Analysis: From Number to Integral. It is fairly short and easy to get through, and will prime your brain for the more intense fare of Rudin's book. It only covers single-variable analysis, however, which is 8 out of 11 chapters in Rudin; most courses in analysis only necessarily cover the first 7 chapters anyways.

I would highly recommend the book by David Brannan called A First Course in Mathematical Analysis (Cambridge University Press). What's good about it is it explains the definitions and walks you through all the "hard points" of the contents. The "other" books are also titled "intro" or "a first" course, but they are actually not so.

Understanding Analysis by S. Abbott is a great introductory text. There is a lot of discussion, both informal to gain intuition and formal to rigorously pin down ideas. It felt like I was having a conversation the first time I read it, which was a very valuable experience as analysis was my first “hard” upper level class.

Edit: I do believe it has a solution manual as well.