You'll have a much easier time at your home institution. You may be able to go away somewhere, but it's getting a bit late in the year to be contacting folks at other universities, especially if you have no experience. Hell, it's getting a bit late even for your home institution.
Keep in mind that money is a huge factor in whether you will land a research gig - most people won't say no to someone who brings their own funds from a research fellowship (hey, it's a free set of hands) but expecting someone to shell out their own lab funds to support you is asking a bit much, especially if you're new to research and don't have much to offer. Deadlines for research fellowships are flying by, however - most I know of have already passed, else will have passed by early March, which doesn't leave you a lot of time to get a research proposal with faculty support together.
Secondly, do you have a specific field of research you're interested, or are you just fishing for a stool and a bench somewhere? PIs will think much, much more of you if you have a specific interest and background in their field, or (even better) a plan of action for what you want to investigate over the summer. That takes some preparatory research on your part, including reading up on the lab publications of whomever you are contacting for research.
At this point, I would talk to people at your school about opportunities, and maybe consider helping out in a lab part-time over the summer just to get some experience under your belt.
