There are hordes of good books in all fields of mathematics. What you need is something you can learn from, not only the best and most glorious of these books.
Books with so many problems and exercises with their hints and solutions are very appealing. But what you really need is a mature and deep grasp of basics and concepts. After all, that's all you need to tackle these exercises with even a surprising level ease and fun.
Analysis is among the most reachable fields in math after high school, and a fair amount of knowledge is required in most of the other fields for beginners.
I do understand the emphasis on solutions because we all deal with self study, at least sometimes, and solutions/hints are crucial to make an evaluation of your own work.
If you are really serious you will soon find out that what you really need are hints not solutions.
Needless to say, hints or solutions are supposed to be a last resort, when there seems to be no way out. Even then, a hint is better taken only partially. And by the way: when tackling problems, it is when there seems to be NO WAY OUT that the actual LEARNING process takes place.
I encourage you to take a deep look into The Trillia Group funded, and free, Zakon's books: Mathematical Analysis I which followed by another volume, but to get some basics, Basic Concepts of Mathematics might be a good place to start.
In the third mentioned book, this was mentioned:
Several years’ class testing led the author to these conclusions:
1 - The earlier such a course is given, the more time is gained in the follow- up courses, be it algebra, analysis or geometry. The longer students are taught “vague analysis”, the harder it becomes to get them used to rigorous proofs and formulations and the harder it is for them to get rid of the misconception that mathematics is just memorizing and manipulating some formulas.
2 - When teaching the course to freshmen, it is advisable to start with Sections 1–7 of Chapter 2, then pass to Chapter 3, leaving Chapter 1 and Sections 8–10 of Chapter 2 for the end. The students should be urged to preread the material to be taught next. (Freshmen must learn to read mathematics by rereading what initially seems “foggy” to them.) The teacher then may confine himself to a brief summary, and devote most of his time to solving as many problems (similar to those assigned) as possible. This is absolutely necessary.
3 - An early and constant use of logical quantifiers (even in the text) is extremely useful. Quantifiers are there to stay in mathematics.
4 - Motivations are necessary and good, provided they are brief and do not use terms that are not yet clear to students.
In the second book, this was mentioned:
Several years’ class testing led us to the following conclusions:
1 - Volume I can be (and was) taught even to sophomores, though they only gradually learn to read and state rigorous arguments. A sophomore often does not even know how to start a proof. The main stumbling block remains the ε, δ-procedure. As a remedy, we provide most exercises with explicit hints, sometimes with almost complete solutions, leaving only tiny “whys” to be answered.
2 - Motivations are good if they are brief and avoid terms not yet known. Diagrams are good if they are simple and appeal to intuition.
3 - Flexibility is a must. One must adapt the course to the level of the class. “Starred” sections are best deferred. (Continuity is not affected.)
4 -“Colloquial” language fails here. We try to keep the exposition rigorous and increasingly concise, but readable.
5 - It is advisable to make the students preread each topic and prepare questions in advance, to be answered in the context of the next lecture.
6 - Some topological ideas (such as compactness in terms of open coverings) are hard on the students. Trial and error led us to emphasize the sequential approach instead (Chapter 4, §6). “Coverings” are treated in Chapter 4, §7 (“starred”).
7 - To students unfamiliar with elements of set theory we recommend our Basic Concepts of Mathematics for supplementary reading. (At Windsor, this text was used for a preparatory first-year one-semester course.) The first two chapters and the first ten sections of Chapter 3 of the present text are actually summaries of the corresponding topics of the author’s Basic Concepts of Mathematics, to which we also relegate such topics as the construction of the real number system, etc.
I did not take these points very seriously, until I started reading and working on it.
It is hard to find yourself completely stuck somewhere: it seems that all have been packed for a person who is learning on his own. Hints are provided whenever needed. In many occasions there are questions like "...Why?" which help in following the text rigorously.