Point set topology

You're right, I think Rudin's Chapter 2 is probably not the best place to first learn point-set topology due to how dense and concise his writing is.

James Munkres' Topology is one of the most common introductions to general topology, and it has some nice pictures in Chapter 2 to give some geometric intuition where topological spaces are first introduced. I like this book. The majority of the exercises are not overly challenging, so it helps to get familiarity with the subject.

I also like Stephen Willards General Topology which is similar to Munkres, but I'd say it's slightly more difficult than Munkres' book.

Finally, although a little older, Kelley's General Topology is a good reference on general/point-set topology, but probably better suited for use after going through some of the previously mentioned books.

If you are looking for a gentle introduction to basic topology ideas with examples and explanations which give some intuition about what is going on, I would recommend that you could have a look at Simmons book "Introduction to Topology and Modern Analysis".

It is available from Amazon here. Some of the reviews on that page are pretty good, and I would not disagree.