Research can require advanced degrees or experience, but it certainly doesn't have to. I got research experience in undergrad simply by speaking to a professor. He wanted summer help with one of his projects and me and two other students of his provided that help. I was still in school, had never done research before, and certainly didn't have any advanced training in what we were studying (it was sort of an ecology project.) Anyways, what I'm saying is that you don't know if you can get the experience until you ask. Also some projects are more intensive than others, so check out what your profs are researching and find something that interests you. Then send them an email.
There are summer research programs for undergrads at the NIH, too, if you're interested. You may have to travel, but I believe they give you a living expense stipend. You can check it out here:
http://www.training.nih.gov/student/ The NIH has a lot of cool training opportunities and they're studying basically everything health related you can think of.
I think a lot of pre-vets who're just starting their experience make the mistake of thinking they need a MILLION hours of experience directly following behind and learning skills from a vet. That's just not true. Any good research experience you get is going to be helpful - even if it's totally unrelated to animals. Likewise, experience cleaning kennels or mucking out stalls is helpful, regardless of whether a vet is involved or even if you get to directly handle the animals much. These little bits of experience can provide you with the stepping stones you need to get more and more or better and better quality stuff.
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Sorry, that wasn't directed just at you, it was more of a general thing.
I'm not sure if getting more experience in your desired specialty is helpful so much as gettibng lots of different experience is. I mean, you don't want to walk into your interviews saying you'd love to do large animal having never really looked at a cow, but you also don't want to box yourself in to, say, shelter medicine now and get lots of that experience when you could be out working with horses or zoo animals or whatever. Basically, you want to get lots of different experience. As many different kinds as possible. That way, you can actually make an informed decision about what you're interested in and, when you get to interviews, you can speak to why you'd like to work in this or that part of the field. Plus, it gives you a broad view of veterinary medicine and all the opportunities provided by it. Definitely a good thing.
For being "well rounded," I'd say just do things that interest you. Don't go out of your way to join organizations just to pad your resume. You'll wear yourself thin participating in activities you have no interest in while getting veterinary and animal and research experience and getting good grades. Just find some things you enjoy and work on those. Leadership positions in clubs probably look better than just being a member, but be sure to only take on what you can handle and keep things in balance. (I say this as someone who's constantly over extended and whose grades suffered because of it.)
That was a little disorganized, but hopefully also kinda helpful. Good luck!