Recommended Reading IST
There's a fabulous, inexpensive Dover text by Alain Robert that gives a very accessible introduction to nonstandard analysis (NSA) through Internal Set Theory (IST). I wish there were an e-book version, but alas there isn't.
There is however an e-book version of another useful accessible book on nonstandard analysis (by Henle and Kleinberg) that comes at the subject from a more "traditional" viewpoint (based on Abraham Robinson's original construction). It's so cheap you have to buy it right now. It spends more time on applications than on the constructions required to define infinitesimals etc.
For someone interested in applied mathematics, the IST version of NSA is extremely accessible, but the techniques are similar in both Robinson's version and IST once you get past the Robinsonian foundations (which require heavy-hitting mathematical logic).
In any case, if you build mathematical models using calculus, or you do statistics, you should know something about NSA because it "algebraifies" analysis. One advantage to that is it brings in lots of possibilities for Computer Algebra. Another advantage to NSA is that it has hugely enriched function spaces. Things that "ought" to be functions but in standard mathematics aren't, like delta functions, are perfectly reasonable nonstandard objects.
In many areas of "standard" mathematics, we have a sequence of "finite" objects which miraculously transforms into a totally different TYPE of object in the limit. In NSA that isn't the case.