There are several problems the Republicans have to address to remain relevant and rebuild their image:
1) Gerrymandering. With huge levels of state control in 2010, the GOP purposely gerrymandered districts to maintain control of the US House despite having millions of votes less than Democrats in House elections. The backlash of this for the GOP is that they gerrymandered the districts to be TOO SAFE. What this means for moderate and reasonable Republicans is that, in a safe red district, their biggest worry is a primary challenge from a candidate to the right of moderate. Consequently, more far-right nut jobs like Ted Cruz are being elected and the moderate Republicans are moving further right to avoid being primaried.
2) "Center-right?": The Republicans like to pretend that the U.S. is a center-right country. They're right that polls show more people identifying as such, but when questioned about *specific* policy matters regarding Medicare, healthcare, Social Security, abortion, gay rights, foreign war, tax policy, corporate personhood, and Wallstreet accountability, the numbers shift overwhelmingly (about 2:1) in favor of Democrats. So basically the Republican establishment has the talk right but the walk incredibly, horribly wrong. Congress has done the Republican brand no favors by being elected in 2010 on a mandate of "jobs, jobs, jobs," and then using their majorities to do nothing but push budget-busting tax cuts for the wealthy, increases taxes on the poor, and the gutting of the arts and social services.
3) Democrats are the new Republicans: I disagree with a lot of Reagan's policies, especially regarding his invention of discretionary public debt and horribly tax policy. But whatever the technical disagreements, Reagan was a uniter, a negotiator, and a leader in every sense of the word. He deserved to be President. His "big tent," however, was only partially his doing. On the opposite side, the Great Society Democrats had veered so far to the left (into actual socialism, not what the tea party pretends is socialism) that they ran off a lot of the center-left. After 12 years under Republicans, Clinton brought the Democratic party radically right (to the actual center), and this is where Obama has kept it. The Republicans now have done to themselves exactly what the Democrats did post-JFK - marginalize themselves while the other side creates unity and concensus.