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What's a good introduction (undergrad level) to modern axiomatic geometry? By that I mean Euclidean geometry, but using a more modern set of axioms such as Hilbert's or Tarski's.

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Elemental Differential Geometry by Christian Bar has a good introduction to the axioms of Euclidean Geometry, although it swiftly moves on to non-Euclidean Geometry (as you would expect). This would be my preferred text for the axioms of Euclidean Geometry and for taking the subject further I would recommend Roger Johnson's Advanced Euclidean Geometry - old-fashioned but very good.