Writing an implementation of a regular expression engine is indeed a quite complex task.
But if you are interested in how to do it, even if you can’t understand enough of the details to actually implement it, I would recommend that you at least look at this article:
Regular Expression Matching Can Be Simple And Fast
(but is slow in Java, Perl, PHP, Python, Ruby, …)
It explains how many of the popular programming languages implement regular expressions in a way that can be very slow for some regular expressions, and explains a slightly different method that is faster. The article includes some details of how the proposed implementation works, including some source code in C. It may be a bit heavy reading if you are just starting to learn regular expressions, but I think it is well worth knowing about the difference between the two approaches.