The opportunities in New Orleans are wonderful, I volunteered with Habitat for Humanity for about a month, and it was an amazing experience to be building houses for Hurricane Katrina Victims that were getting things back together. We slept at the local YMCA and just paid for gas, which ended up being around $30 per person, so it turned out to be an extremely affordable program. My entire living expenses for the month was probably $200, and that was all leisure expenses, meals were provided by the YMCA.
But as far as volunteering abroad, I hear it can get costly!
If you're interested, look up Alternative Spring Break. You will basically spend a week helping out a community in some pre-determined city across the country. Also, check with your study abroad office to see if there is any medical trip to a Central American country. Those are highly rewarding.
My best study abroad experience was with a language school: www.ecela.com . I spend just shy of two semesters there, was pretty fluent when I came back, which has helped a good bit on the clinical part of medical school.
I'm studying abroad in Spain right now, though I'm not doing a volunteer program or anything like that. I'm a Spanish major and I'm in Oviedo (northern Spain) for a semester to study Spanish. I'm not sure if this was the kind of experience you were looking to have, but it's been great for me and I'm really glad I had the chance to do it. It's been great living with a host family, speaking mostly Spanish and eating a lot (I have to take advantage of the opportunity and make the most of my time here - they say "Hay que aprovechar" which means exactly that.)
I can second the value of a language course abroad. I spent a summer at the Goethe Institut in Mannheim, Germany and it was a great experience. I became fluent in German, met some friends I still have 10 years later, and learned a lot about the world. My only regret is not going for a longer stay!