I would like to write a program that will find bus stop times and update my personal webpage accordingly.
If I were to do this manually I would
Since you write in C, you may want to check out cURL; in particular, take a look at libcurl. It's great.
You can use the mechanize library that is available for Python http://wwwsearch.sourceforge.net/mechanize/
This is called Web scraping, and it even has its own Wikipedia article where you can find more information.
Also, you might find more details in this SO discussion.
Beautiful Soup is a Python library designed for parsing web pages. Between it and urllib2 (urllib.request in Python 3) you should be able to figure out what you need.
What you're asking about is called "web scraping." I'm sure if you google around you'll find some stuff, but the core notion is that you want to open a connection to the website, slurp in the HTML, parse it and identify the chunks you want.
The Python Wiki has a good lot of stuff on this.
You can use Perl to help you complete your task.
use strict;
use LWP;
my $browser = LWP::UserAgent->new;
my $responce = $browser->get("http://google.com");
print $responce->content;
Your responce object can tell you if it suceeded as well as returning the content of the page.You can also use this same library to post to a page.
Here is some documentation. http://metacpan.org/pod/LWP::UserAgent