

1. Repeal Obamacare
This is a very popular idea and one they will almost certainly do. However, Obama will almost certainly veto it. Then what next? The most effective thing they could do is to block bills that fund it etc doing what they can to delay any further implementation until after the 2016 election.
2. Avoid unforced errors
You know the stupid stuff that is politics 101. Now that they have majorities in both Houses in Congress, the left leaning media will be out for blood. They will have enough problems without doing stupid things.
3. Make a big point of paying attention to their home districts
Eric Cantor is a lesson that shouldn't be forgotten soon. There are NO safe districts for Republicans anymore. Moderate districts are in danger from Democratic challengers, conservative districts are in danger from Tea Party challengers. They need to pay attention to the fundamentals of politics 101, the voters elect you. They need to at least do a convincing job of pretending to care about them.
4. Pass the Keystone pipeline
There is enough Democratic support for this that with careful negotiation and compromises Republicans can form a veto proof majority. Democrats from oil states and union states have publicly supported Keystone in support of #3 in their own districts, those Democrats want a bipartisan bill to show their own constituents and are willing to work with the Republicans on this.
The opposition comes from the environmentalists in a true example of it sounds good it must be good regardless of whether or not it actually is good. They are blocking the pipeline in the belief that blocking it will magically make that nasty oil go away. What it has done in real life is to shift it rail transport, which to date has been far worse for the environment (spills, accidents etc) than a pipeline would be.
5. Block the EPA from issuing carbon regulations.
They actually don't have the authority to do this and are wasting government money getting sued. There is so much Democratic opposition from union states and coal states to these carbon regulations that there is 0% chance that the authority they need will actually pass congress under either Republican or Democratic control. So the Republicans should get together with the Democratic opposition and form a veto proof majority to tell the EPA to stop exceeding their authority and wasting money we don't have.
6. Don't touch comprehensive immigration reform with a 10 foot pole!
No matter what they do here, people are going to hate it. They'd be better off pretending to work on comprehensive immigration reforms, while actually only passing small immigration reform bills. Frankly I think comprehensive immigration reform is doomed to Obamacare overcomplexity failure syndrome and multiple small bills will probably be more effective anyway.
7. Fast track authority for trade deals.
That is currently stalled in congress and for good reason. They don't trust Obama's state department to effectively negotiate a decent trade deal with anyone. I agree entirely and in fact I would expand it from just Obama's appointees to State as a whole. Their past performance on trade deals under both Republicans and Democrats has resulted in the current outsourcing jobs crisis.
However a good trade deal could help the economy. So Republicans could work a deal with Obama to select someone outside of the Washington establishment to negotiate. Trade really isn't the area of expertise of politicians, they need an experienced, highly effective businessman that Obama would accept as an envoy. Warren Buffet or Bill Gates perhaps?
8. Vote down every single left wing judge Obama appoints
The judiciary is key to the future of the Republicans. Obama loves to appoint far left liberals. They Senate should vote every last one of them down. If by some miracle he actually appoints a moderate, they should vote that person up. By voting in favor of moderates but against liberals, they can make a very convincing case that they are not the party of no but merely want balance.
9. Reign in the power of the Consumer Financial Protection Bureau
The legislation that created this created a monster without checks and balances. It's new so that the power hasn't been abused yet... But this is Washington, it's only a matter of time. Congress needs to get those checks and balances in place now! Careful negotiation with Democratic Senators should be able to form a veto proof majority on this issue.
10. Pass the Regulations from the Executive in Need of Scrutiny (REINS) Act
This is another check and balance situation. Basically any regulation the executive comes up with that has an impact of $100 million or above needs congressional approval (yes or no, we approve or we don't).
Note that bill should be inflation adjusted to keep things from getting out of hand. Congress doesn't need to approve every little thing.
11. Work on easy regulatory reform
Regulations that are completely ineffective or outdated provide a compliance burden on US business an thus are an unneeded drag on the economy (and jobs). There are laundry lists of these regulations/laws all over the place. Every think tank has one or more. The GOP should take the easy win and get rid of these things. Most of them have small impacts, but each one adds up and frankly the economy could use all the help it can get! They might even be able to get a bipartisan veto proof majority.
12. Pass a budget and pass it on time
The democratically controlled congress could never seem to get this done. If the GOP wants to demonstrate competence, basics such as this are a good place to start. Of course Obama will veto anything they come up with, but they should do their job anyway.
13. Corporate Tax reform
There is support for this on both sides of the aisle as it should help jobs. Negotiate with Democratic Senators to form a veto proof majority in order to remove corporate tax loopholes put in over the years to benefit the crony capitalists, while lowering the nominal rate. Any bill should be designed to be revenue neutral.