From the OSU Vet website (the program I am most familiar with): A minimum of 80 hours of veterinary experience are required to have your application considered for admission; however, admitted applicants have more than a 1,000 hours of quality "hands-on" experience in the veterinary field. You are expected to have a realistic and appropriate perspective of the responsibilities of the veterinary profession. Your experience may come from a job or volunteer service. You may acquire experience by working with veterinarians in private practice, farms, ranches, animal shelters, zoos, aquaria, laboratories, etc.
Visit UC Davis site and look it up! Every school is different. Keep in mind most vets are not going to pay you to shadow. We are further ahead to hire someone who is available 20-40 hours a week year round. Also, liability issues come into play with volunteers. In addition, you will see more of what you want to see if you are volunteering vs a paid position because you have the flexibility to float to see what you want within a clinic. However, it is tough to get in with a clinic to shadow. We have tons of kids ask every year and can only take so many on. We tend to take them on by seniority.....current vet students get first dibs then on down the line. You can also get experience by pet sitting, volunteering at the humane society, working at a pet store, working at a groomers as well as the suggestions above. Join FFA or 4-H!
Also, if you if get the ok to shadow...be smart about it. Dress appropriately (no jeans with holes or that leave body parts stick out, cover the tattoos, no low cut tops, easy on the make up, pull out the piercings. Leave the gum at home. Stay off the cell phone while there. Be quiet in the exam room...it is the clients time. Remember you are representing the clinic. Be on time. Don't touch the animals unless told it is ok (think liability). It is ok to ask some questions but remember I am there to work and not just to entertain you! As you earn my trust and confidence, I will let you do more. Some kids have been so disruptive to the flow of the clinic, we don't ask them back. Oh, and remember, we can't save every pet. And if you do ultimately get into vet school....pay it forward when someone else asks! Sometimes having someone hot on your heels all day really slows you down and it is not the most convenient thing to do but do it anyhow for someone else someday!!!