Amateur radio (ARES and/or RACES) or the American Red Cross are two great places to get started. Most Red Cross chapters actively respond to house fires where the residents need assistance, so their volunteers get experience year-round. A lot of amateur radio groups help provide communications for events like fundraising walks/rides, parades, etc. which helps volunteers learn to communicate effectively during chaos. CERT is another great organization to get involved in but it really is just for your local community.

I don't think any of the larger organizations would ever send a "green" volunteer overseas to a disaster. There are plenty of para-church organizations that would, especially if you can pay for your transportation.