To the OP, don't underestimate shelter experience! It's where I got my foot in the door, definitely. Almost all shelters are looking for people to help, and a lot of them will let you be hands-on with the animals if that's what you WANT to do. I currently volunteer at a no-kill shelter, and I get to walk dogs, bathe dogs, feed the animals, plus other things like clean cages and wash bedding. So much volunteering has allowed me to get a position with a local SA clinic, which is just shadowing right now. However, it will progress into more hands-on things as the vet and I get more comfortable with each other. So, my advice is to volunteer at the shelter as much as you can. Create a resume and cover letter, making sure to put your volunteer experience on there, and start sending it around to vets. Call them to follow-up on it and let them know how interested you are in the profession.
Also, somebody mentioned the zoo. I will be doing a 10-week, unpaid internship at the zoo this summer. I get to assist the veterinarians in the onsite animal hospital. I found out about this opportunity by checking out the website for my local zoo. Most zoos have these, so that's another option for you to consider. If you already have some volunteering experience, it would be awesome on your internship application.
Good luck!