Hi OP.
There are a lot of different programs that you can look into. It all depends on what exactly you are looking for.
I'm from a CA school most of the UCs (University of California) [though I suspect most US campuses have chapters set up] have chapters set up of international organizations such as Global Medical Training (GMT), Foundation for the International Medical Relief for Children (FIMRC), Global Medical Brigrade (GMB) and Flying Sams. These programs are good if 1.) You don't want to be too far removed from amenities like running water (though it was often not heated nor good pressure), a toilet, a bed and food provided, 2.) You also want to experience the local culture (I guess previous posters said its like a little vacation also) 3.) You want to work with local doctors rather than doctors from the US 4.) You want to be around people your age and make friends (and have fun too, aka a little vacation). Look at your local school or look into other schools contact them and let them know you want to join if this seems like an option for you.
If you want a different experience you can look into private religious organizations that go abroad. Such as Tzu Chi Fondation (Buddhist) or a Christian organization. Though I am not fond or religious missions, look into it many times they do not put religion into what they are doing. And actually a lot of times, for Christian organizations anyways, they work through local churches to set up the health clinic and going to mass/sermons are optional. Note when you go with these organizations you will be going with US doctors but who are all friendly and they do want you to help & give you a great experience (my friend was able to stitch up a patient after a surgery!). But you will not be with people your age...
The organizations I mentioned above at schools typically don't do surgery they do health clinics but I still found it well worth it as you will rarely see these ailments in the US (ie parasitic infections and work related problems). A good thing about FIMRC is that they work to establish long-term clinics there so you won't only be seeing patients you are helping to build a clinic and there are also non-medical duties. Another great thing about FIMRC that I like is that they take days to actually teach health practices to the local children and the expecting mothers so that they can avoid re-occurances of medical problems.
I recently went with GMT for a week to Nicaragua it was an amazing experience but I did have to pay 1000$ + airfare. So for a week I paid ~1600$ and was wary of the fee but I think it was well worth it, I do speak basic Spanish but I can understand well so I was able to converse with patients; the experience helped me greatly improve my medical Spanish (which has helped me back at my medical office job in the US). The doctors there basically let you be in charge. You are with the patient and a translator; you get to tell your diagnosis and the treatment needed, the doctor will then come in and tell you if you are right or wrong. Its a great learning experience to build your patient-doctor skills, which I feel US volunteering lacks. You also learn to recognize things that you normally wouldn't realize.
About the fees you need to understand that the fees for volunteering do not only go to room and board. In the countries that you are going to the translators and some of the doctors they don't have the opportunity to take off a lot of time, every day working is necessary for them; so many times the company is paying for translators, doctors to head the programs, organizers to contact cities of interest and set up camps and drivers to transport you here and there (GMT, FIMRC are a year-round missions, for the local people who work the missions this is typically their only job). In addition to paying for this you are also paying for medical supplies. It would be pointless to go to a rural area and say to them "Your child has iron deficiency you need to go the local clinic _____ miles away and buy XXXX amount of iron pills for XXXX dollars." Instead what happens is you tell them what they have and give them medicine for it. You will be seeing up to a hundred patients a day and thats a lot of medicine (multiply that by how many clinics per week). GMT gave at least a month supply of medicine to each patient. But I def. suggest looking into where your money is going to. E-mail or call them and ask where the money goes too, and if you get no reply like a previous poster its better to avoid!
I hope this helps you out! It all depends on what you are looking for and what you have money for. Good luck, i've done a lot of international relief and its totally different from the US but its always what you make out of it!
Oh and also some of that money went to non-medical activities. We went zip-lining and took tours of the cities and went to museums and spent the day at a lagoon. Some of these "missions" do allow a little fun, you get a little fun vacation out of it; if you can afford it anyways..