The first analysis course I ever took used the book "Elementary Analysis" by Ross. It's basically baby-baby-Rudin. Ross's book (if you include the exercises and the optional sections) covers more or less the same material as the first 8 chapters of baby-Rudin, but the exposition is much friendlier and it's more easy-going for a beginner. When I say "beginner" here I really mean beginner -- someone who has never even written a rigorous mathematical proof. The book would probably be very boring and tedious for someone above this level.
Kevin H. Lin
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