'SOMEWHAT VICTORIES'

But even some of the most conservative House members also were expressing the need to ease their hard-line stances.

"It might be that I end up voting for some things that are not clear victories but they are somewhat victories. But that's what the legislative process is about," said Fleming, who helped orchestrate the 2013 federal government shutdown in a failed attempt to kill Obamacare.

Republicans still may not be able to achieve a full repeal of Obama's signature health care law, but they are expected to chip away at it by passing measures to ease requirements for employer healthcare coverage and repeal an excise tax on medical devices. On the energy front, the first major bill the Republican-controlled Senate intends to pass is approval of the Canada-to-Texas Keystone XL oil pipeline.

In their first vote on Tuesday, the Republicans muscled through a controversial change to rules for estimating the cost of four major tax and budget bills in the House, to include revenue from anticipated growth effects. Democrats decried adoption of so-called "dynamic scoring" as masking the true costs for tax rate cuts that Republicans intend to pass this year.

Another big test of Republican leadership is likely to come by mid-year, when Congress will either have to raise the government's borrowing authority or risk a credit default. The debt limit fight is always a hot-button issue for conservatives who oppose more borrowing.