WASHINGTON—The nation’s new, all-Republican leadership begins to take power Tuesday with an ambitious agenda of tax cuts, regulation rollbacks and repeal of President Barack Obama’s health law, but they face a complicated legislative path pocked with unresolved policy details.

After new senators are sworn in Tuesday by Vice President Joe Biden, a symbol of the departing Democratic administration, and House members by GOP Speaker Paul Ryan, the Republicans among them plan to move quickly to turn back an era defined by Mr. Obama’s...