Why wouldn't Mozilla work, what do you mean by that?
Mozilla is decent, but bloated; that's fine though, if you like and use those added features. I tried it for a while, but I found that I didn't lke their IRC and Mail clients. Mozilla development has stopped though, so it's not likely to get any updates aside from security patches.
Firefox is just the slim version of Mozilla; it's just the browser. It's being focused upon for development, so it will continue to get better over time. It's also less bloated and more customizable.
Other than that, you have a few IE shells to pick from: MyIE2, Slim Browser, Avant etc. They only add functionality to IE though, they do nothing to enhance it's security.
If you're not sure, just try a few until you find what you like.