In my view, the best place to start for a good grounding in rigorous mathematics is Velleman's 'How to Prove It'
As for a good broad overview of many areas, the title that springs to mind is 'the nature of mathematical modelling' by Gershenfeld, though you'd better have some decent maths experience before tackling that one - it can be tough-going, but is refreshing in its breadth and clarity.
As for a good broad overview of many areas, the title that springs to mind is 'the nature of mathematical modelling' by Gershenfeld, though you'd better have some decent maths experience before tackling that one - it can be tough-going, but is refreshing in its breadth and clarity.