Elephant Butte has some great fly fishing for bass. Most people fish the lake by trolling for striper or casting plastics into the coves for largemouth bass, however, that's not to say that the lake doesn't offer some great fly fishing. In terms of warmwater fly fishing, Elephant Butte has the best fishing that NM has to offer. I would head out to the lake with a kayak or a boat and cast out streamers - there's a good chance that you'll pick up some nice fish. I would fish with a nine weight rod and a high quality reel with smooth drag. Make sure you have a good fly fishing reel! You'll need a good drag to stop a big striped bass. My favorite fly for the lake is a clouser's minnow. In the evenings you might get some good topwater action so have some topwater flies ready.
Not directly related to fly fishing... but this is a pretty good article about tips and techniques to use when fishing for largemouth bass.