The most clearly written is John Baylis's 'What Is Mathematical Analysis?'. I _strongly_ recommend you read this. It's 125 pages of clarity and intuition building.
More rigorous alternatives would be: 1. Elias M. Stein's 'Real Analysis: Measure Theory, Integration, and Hilbert Spaces'. 2. Robert S. Strichartz 'The Way of Analysis'
Personally I would use the Baylis book and the other two as reference.
Additional resources include the top voted mathoverflow.com and math.stackexchange.com answers. Beyond useful.
Lastly, I have a twitter account (@math_twitr) that indexes (mostly) academic mathematicians. You might want to look through my follows there, they regularly post useful materials.
Addendum: Hm, in addition, I think you might want to look through this Amazon wishlist of mine consisting of those math books recommended for clarity by those who should know: https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=de...