For the last week since the election, the overarching message to the Republican Party has been: "ADAPT OR DIE". There have been numerous articles and essays showing how demographic trends are going to make it more and more difficult for the party of old, white, Christian men to be relevant in 21st-century America. Already, you're seeing a partial understanding of this as the Establishment faction of the party pivots on immigration reform. But you're also seeing a retrenchment among the most hardcore social conservatives, invoking the tried-and-true(-and-failed) refrain of "We lost because we weren't conservative enough!"
So...grab some popcorn and stay tuned in for the next couple of years because you're in for some fireworks as the GOP gears up for a civil war in its ranks between the small but still extant Moderate Wing, the Establishment Wing and the God Wing.
If the Moderates win, the GOP could be back as a relevant challenger and peel off support among conservative and pragmatic Dems, even as the centerpoint of the mainstream skews left.
If the Establishment wins, the GOP will still be a rump party for a while but I think will try to become broadly pro-Hispanic and rely on the strong correlation between Hispanics and Catholicism to bolster support for some social conservative issues like abortion and gay marriage. They'll lose a lot of the nativists, old-school racists and rabid Protestant Bible-beaters, but then there's not much of a place for those folks to go--might jump to a further-right party like the Constitution Party.
If the God wing wins....the GOP could be headed to the political dustbin within a couple of decades, to join the Whigs, Prohibitionists and Free Soilers.
My personal hope is either for the 1st or 3rd outcome. The former because a balanced, rational party of fiscal conservatism and socially laissez-faire would be a welcome thing in the US political system and a good counterbalance to the inevitable excesses of a long period of Democratic dominance. The latter because nature abhors a vacuum -- if the GOP closed up shop (or withered to the point of irrelevance), the Dems would eventually splinter (hopefully into more than a two-party paradigm).