So what do the Republican do now?

I suppose number one duty is to prevent the lame duck session of Congress, that will get going shortly, from doing any more damage. Despite the election, the old Congress remains in power until next year. The democrats will surely call a lame duck session to pass as much stuff as they can before next year gets here. Cap and tax, card check, appointment of democrats to every vacancy, more Obamacare, tax hikes, and who knows what else. Hopefully the Senate Republicans will be able to stop all or most of it.
Then next year, with a solid Republican House, what should the Republicans do? The house can pass anything it likes, but the Senate Democrats will be able to stall it, and the President can veto it. The Republicans have to convince the country that they are serious about cutting spending and getting the economy moving if they want to win in 2012. If the Republicans just vote thru things in the House and have them die in the Senate, or get vetoed, the public ain't gonna like it. And they will remember in November. The public doesn't care why nothing happens, they will turn the rascals out in 2012 just for doing nothing.
So what can a Republican house do that works, given a hostile Senate and executive?
For openers, they could vote down pork. For instance, the periodic "Farm Bills" lavish our tax money on farms largely owned by corporations. There are few family farms left, and not enough family farm voters to make it worth the cost of the farm bills. Same goes for the "Highway Bills" that benefit road contractors and real estate interests, and the "Energy Bills" which subsidize the oil and gas industry.
Tighten up, or even better, kill entirely, earmarks. Congressmen love earmarks because they act as bribes or payoffs to supporters. Voters hate them because it's pure waste money. Voters are more important to re election than paid off supporters.
Refuse to pass omnibus spending bills. Each executive department (Defense, State, HHS, Energy, and so on) get one, separate, budget bill. If the bill fails to pass, that department can no longer write checks. This way the voters and the newsies have some idea of how much money gets spent on what. An omnibus spending bill for the entire federal government is so damn big that nobody can figure out how much is going where. Break it up into smaller pieces and it becomes manageable. Divide and conquer still works.
Reduce appropriations for economy killing agencies like EPA. EPA was created to clean up airborne smog. That's been accomplished, the skies over our cities are blue again. They used to be yellow. Now EPA is full of bureaucrats trying to stay employed by making red tape to hinder economic growth.
Find some utterly worthless government operation and shut it down completely, lay off all the employees, sell the office, and burn the files. That will generate a flood of favorable press, AND scare the bejesus out of the rest of the bureaucrats. A scared bureaucrat is a good thing.