How to reproject a raster in QGIS?

I am a history teacher and I like to make my own maps (I can never find exactly what I want). I am using QGIS 1.8 and Natural Earth data.

I want to reproject their cross-blended hypso rasters into projections that are more familiar looking to my students, like Lambert Conic Conformal.

I have tried changing the crs for the whole project, and the warp and assign functions from the raster menu, and changing the crs for the raster layer itself. None of these seem to work properly - the resulting maps are unrecognizable and extremely large (file size).

I feel like I am making some kind of basic mistake but I don't know where to begin with this. Any help would be appreciated.