In terms of religious demographics, the Republican Party would be wise, over the long-term, to reach out to at least
some of the members of the fastest-growing religious group in the American population-the "Nones." (Note: this group is NOT the same as atheists/agnostics, who IIRC, are pretty overwhelmingly Dem now-for understandable reasons, of course-but are too small a group to make much of an impact). How they do that, I'm not sure.
Beyond that, the GOP needs to improve its margins among Catholics (and not just white non-Hispanic Catholics...) and mainline Protestants. A party that is overwhelmingly dominated by White evangelical Protestants in general, because of the way that many evangelicals'(particularly the political ones
) aggressive and frankly, obnoxious style turns off non-evangelicals, will not be able to reach out to much of the population. Furthermore, the dogmatic, rigid beliefs of many evangelical Protestants (and the hypocrisy of many megachurch pastors
) also turn people away from an increasingly evangelical-dominated Republican Party. Finally, white evangelicals are declining as a share of the electorate.
Beyond this, I'm not sure what else the GOP can do. It's in a position where it needs to turn out both "the base" in large numbers while at the same time, not alienate "swing" voters.