Calculus by Michael Spivak is a very readable introduction to real analysis-- that is, analysis on R.
If you want something which starts with R and moves onto the more general metric space, Hilbert and Banach space setting though, a very readable (Dover) book is Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
You might also appreciate a book on topology if you're looking for something of the latter category-- I'm currently reading through Introduction to Topology by Theodore W. Gamelin and Robert Everist Greene.
Both books were recommended to me in this thread, so you might want to take a look there as well.