I agree with all of the ideas above. In addition:
If you are going to be volunteering at an animal shelter, ask the vet there to shadow him/her.
Or volunteer for a summer at a shelter and use that experience to build your resume then apply for jobs as a kennel worker or vet/tech assistant. That's basically what I did, except I volunteered at a zoo for 2 years, worked at a petting zoo one summer and used that experience to get an assistant position. I then used the assistant experience to get a tech job at an ER (I live in a state that doesn't mandate LVT's).
Basically, you have to think long term and be prepared to do some nasty work (ex: shoveling bear poo). Good luck