Don't use galvanized- it comes with a pretty nasty zinc coating that is a pain to deal with.
As for jump rings/wire, you can look for aluminum- over time it will get a funky coating, so anodized aluminum is preferable. Also stainless steel holds up well. If you don't mind a patina over time, copper is an option. Otherwise you can go with brass.http://theringlord.com/cart/
sells cool chainmaille supplies both wire and rings. They do carry a variety of materials like those I listed above, and they come in pretty much whatever gauge they want.
What sort of thing are you going to make? Give me a little more description and I might be able to line you out a bit more.
If it doesn't have to be gorgeous, you can look for a thin gauge of aircraft cable. That lasts really well, and is super strong. You can do a lot with it, with only a few special tools - bolt cutters and some crimpers.
Depending on what it is, you can find some different ball chains or other chain made for hanging light fixtures. Some of it is quite nice looking. They can be found at Home Depot or your local small hardware store. (Go to one of the small local ones and describe what you want, they can usually help out WAY more than the box stores.) Basically things that are OK in plumbing, or light fixtures made or damp locations should work!