At issue immediately is whether that can be achieved through a shift in tactics and tone, or will instead require a deeper rethinking of the party’s longtime positions on bedrock issues like guns and immigration. President Obama intends to test the willingness of Republicans to bend on those issues in the first months of his new term, when he plans to push for stricter gun control and a comprehensive immigration overhaul…

“Republicans will get their mojo back when they define themselves as the party of economic growth and upward mobility,” said Gov. Mitch Daniels of Indiana, a Republican who will become the president of Purdue University next week. Mr. Daniels said new lawmakers and governors — many of whom are minorities and women — would reshape the Republican Party…

Some are talking about the need to find a positive vision and agenda that represents conservative values but still speaks more directly to the concerns of a broad section of voters — and manages to sell that vision through leaders who can convince voters that the party wants to move forward and not back.

Former Senator Kay Bailey Hutchison, a Texas Republican who retired this year, said Republicans must shift their focus away from issues like abortion, same-sex marriage, gun rights and immigration.