My three main priorities are that it's readable, not particularly dense, it has problem sets, but also very importantly I'd like for it to contain or have a solution set available for me to check if my answers and intuitions are in fact correct.
I'd also like something where I can use my visual intuitions.
I was told Analysis in Euclidean Space by Kenneth Hoffman was good but I can't seem to find a solution set available.
On
I really liked Paul Sally's "Fundamentals of Mathematical Analysis (Pure and Applied Undergraduate Texts)." The book is highly theoretical and doesn't have any computational problems, but it is super insightful and incredibly challenging. The independent projects and legitimately challenging exercises which are posed provide an incredibly good analysis learning experience.
On
I recommend Robert Ash's Real Variables with Basic Metric Space Topology. The book is cheap (new paperback copies are sold at USD 11.95 each on Amazon right now; a free and legal digital copy is also downloadable from Professor Ash's website). The exposition is clear, efficient and user friendly. The book is thin (only 224 pages), but its coverage is just right for a first course (there are other user-friendly introductory texts on analysis, such as Victor Bryant's Yet Another Introduction to Analysis or Lara Alcock's How to think about analysis, but I believe they are too elementary). It is quite possible to finish reading the first six chapters (up to Riemann integration, 116 pages in total) within two or three weeks. The book has many pictures and there are also solutions to selected problems.
You would probably want to check out some of the books from the Springer Undergraduate Mathematics Series (SUMS). I own several of them and they seem to be well suited for self study and they have solutions at the end of each book.
Since you don't specify what you mean by "Analysis", I can't be any more specific than this, but the series has books on Real Analysis, Complex Analysis, Probability, Functional Analysis, etc.
If what you're looking for is an introduction to rigorous calculus with the usual $\varepsilon$-$\delta$ approach, I would recommend you to take a look at Stephen Abbott's "Understanding Analysis" and Kenneth A. Ross' "Elementary Analysis: The Theory of Calculus", both on their second editions. Ross' book has solutions at the end, and while Abbott's book doesn't, his explanations are really good.