>>863
A gentle intro will not be very beneficial and will waste time. The reason analysis is hard is because the ideas are presented in a way that is isolated and general. The same will go for algebra and topology. The books on those subjects should be read, but not be the center of your study. They are there for reference when needed.
I studied analysis and ignored everything else and it failed me. I then changed my approach, got a book on differential geometry (lee) and began to read that, referencing books on analysis, algebra, geometry and topology when it was necessary. Math is a slow process and the time you put in will significantly outweigh the knowledge you gain.
Also a little side note, get into complex analysis asap. Both (real and complex) deal with the same ideas, just complex analysis is more tangible due to its geometric nature. I recommend Ahlfors book.