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 

Visit www.calgarytransit.com
Enter a stop number. ie) 9510
Click the button "next bus"

The results may look like the following:

10:16p    Route 154
  10:46p Route 154
  11:32p    Route 154

Once I've grabbed the time and routes then I will update my webpage accordingly. 
I have no idea where to start. I know diddly squat about web programming but can write some C and Python. What are some topics/libraries I could look into?
That site doesnt offer an API for you to be able to get the appropriate data that you need. In that case you'll need to parse the actual HTML page returned by, for example, a CURL request .
Grabbing text from a webpage