Where to begin studying Biology in earnest, with preferably free sources?
Before getting into veterinary and mammalogy, you might want to start with the basics. As many people have mentioned, Campbell and Reece is one of the best textbooks to start a foundation in biology. If you live near a public library you might find a 6th or 7th edition to borrow. Amazon has a few 7th editions for cheap as well. Although it is an introductory biology book, it will give you a wonderful tour of cellular and molecular biology that will make it much easier when studying animal physiology (which they also covered in the later chapters to a limited extent). 

Free Biology Text Books:

Boundless Biology: Boundless is a company that has many free textbooks for high school level biology to help teachers in have a free online textbook. 

I have also found a list of free biology text books here that might be of use.

Free Mammology Textbook:

As part of a National Science Foundation Grant, the Northwest Center for Sustainable Resources created *Introduction to Mammology*. It is a free textbook that was developed for a two-year Wildlife Technology program. It also contains information related to the lab portion of the program, but you can skip those and focus on the material. 

Free Online Coursework:

Coursera: A few people have already mentioned Coursera, which is a Massive Open Online Content (MOOC). Here, you can find many courses across a wide range of biology topics (and others!). You can enroll in a course (for free) and many times they come with accompanying free textbooks as well (at least that is what I have experienced).

edX: Another MOOC that was started by Harvard.

Khan Academy: Another great website for explanations of certain aspects of biology. As Tyto alba stated, there are some errors in the videos.