Mathematical rigor is not the most important thing when first learning strings, there are many things that are not possible to formulate rigorously, because the best language for doing this isn't known. In addition to Polchinsky (which is excellent), I recommend reading Green Schwarz and Witten, and also the original papers, since these have points of view which are not found in later articles, but are profound and important.
These are found in two very good reprint volumes: "Dual Models", and "Superstrings" (although much of vol II is well covered in Green Schwarz and Witten and Polchinsky). These are essential for properly understanding the subject, even today. The issue is the Regge ideas and the S-matrix ideas which are glossed over in later treatments.