What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
I wouldn't know as it has been over 10 years since I looked at a math textbook. The thing about math is it doesn't change. Well at least the basics so it's safe to say any text that you find in the library would be a good source.
Where different text books may deviate from one another is how they prove a theorem. Like programming, you can usually get the same results by going down different paths. Some paths are more efficient than others, but that is predicated by what you know.
If you are just learning, the best thing to do is find textbooks with answer keys to assignments. Also with the advent of google and such, I would have to imagine you can probably find answers to a lot of the questions that would be posed in these text books so answer keys may not be all that important now.
I quite liked Stewart's Calculus. http://www.stewartcalculus.com/
And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.
If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 is greater than 0, you'll learn to look at things differently.
In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.