If you want to go backpacking and hiking, as opposed to just budget traveling and sightseeing, I recommend New Zealand. It's been years since I've been there, but I lived there for a while as well and in general, they like Americans, as long as you aren't some obnoxious yahoo who waves the flag and finds fault with everyone else on the planet, like some American tourists do.

There is crime everywhere, but some places are worse than others. One of our members had all his gear stolen in Hawaii. I lived there for years and that place is really bad for property theft. I wouldn't walk away from anything there I wanted to ever see again.

I also lived in South America, but it was a long time ago. You need to be aware of the political situation in each country you travel to. Peru shouldn't be too bad. I'd stay away from Venezuela and Bolivia (where I lived). The political situation in both is anti-American for the most part. I'd definitely avoid Columbia-there is an ongoing insurgency that has been going on for years and it is a major drug trafficking country.