StackOverflow is the de facto standard place to go when you need to figure out how to do something as a programmer. A web search for information about how to check the number of lines in a file using the C programming language will probably give you StackOverflow questions as the most relevant search results. Unfortunately, while writing good, clean, reliable C does not need to be difficult, it is certainly difficult to find information about how to do it well, and the people answering these questions on StackOverflow are not filtered by their skills or knowledge of good practices. From time to time, the top-rated answer or the accepted answer is even the worst answer that addresses the question.
The best example I have seen of a straightforward line count algorithm is in a place that might seem obvious after the fact: the source code of the wc command line utility. Figure out how the -l option is handled in a good implementation of wc, and all that's left is to adapt it to your needs. Ideally, the solution you find should be short, simple, careful (e.g. checking for error conditions), robust, well-worn, and well-formatted.
The C coding standards of the OpenBSD community are among the highest and most pragmatic you'll ever find, and when someone makes a decision about implementation whose reasoning is not pretty obvious in the code itself you're almost certain to see that reasoning clearly documented in code comments, which can also teach you something about being a good programmer in general.