This is a good opportunity to emphasize the critical status of starting with unambiguous definitions as the first step of any proper mathematical presentation (*). Be sure to provide them with the definitions as stated in the textbook that you're using. Then, granted that students may not be accustomed to reading with such laser-intense care to each word in a definition, it's likely important to run a rather large series of exercises to emphasize that each word of the definitions is important, and to internalize the meaning of each type of number.
This needs to be an "automatic" skill for students, such that they can instantaneously recognize each type of number later on. I would recommend the timed quiz site that I developed so as to exercise exactly this topic (and others like it):
Automatic Algebra: Sets of Numbers
In addition, look for opportunities throughout the course later on to use these definitions and reflect on them again. For example: Perhaps we allow exponents $a^n$ where the base $a$ is real and the exponent $n$ is an integer. Review: What does that mean? Define radicals $\sqrt[n]{a}$ where the index $n$ is natural and the radicand $a$ is real. What does that mean? Define polynomials as a sum of terms $ax^n$ where the coefficient $a$ is real and the power $n$ is natural or zero. What does that mean? Run a series of introductory exercises in each case where students must identify whether expressions qualify for each definition or not.
Overall, this subject is among your best chances to highlight the importance of careful definitions, and how they will be used to define and illuminate later concepts throughout the course.
(*) Fine print: It should go without saying that any presentation starts with some undefined intuitive terms; in this case it's the natural numbers. But integers and rationals can then be rigorously defined, and this is a prime opportunity to practice the axiomatic thought process in a basic algebra class. Real numbers will be somewhat hand-waved at this level.