Cornell sounds more impressive than U of F (no offense to any U of F grads out there) Cornell just has that certain snooty coolness to it
Start as Pre-vet, and get your undergrad degree in that, then you gotta go to vet med school, which is grad school, as latazyo said. If you get this undergraduate degree, and have good grades, (and do well on your GRE's, which are like SAT's to get into graduate school) then you probably wont HAVE to have work experience to get accepted into the grad program. This however, is a question you can get answered with more certainty from your student advisor while you are still an undergrad.
In terms of getting a job post-education, You dont HAVE to work in a vet's office to land one, but nowadays a lot of places hiring look most favorably upon candidates who have both the necessary sheepskins AND related work experience.
Plus you gotta pay the insane tuition somehow, right!
Congrats on graduating HS! College rules.... Just remember, its all about balance, so dont go too crazy freshman year!