Thus far into Obama's presidency, the Republicans have done very well by being obstructionist. Obama extends his hand and offers to work together, and they both slap it aside and call him divisive. Congressional Democrats adapt Republican ideas like cap and trade and the public option and the Republicans scream Nazi socialism. They haven't had a single positive idea, and for them that's a good thing. A positive plan would just confuse their supporters.
So, when the Republicans win the House, they will just continue to do what they've been doing: Demonize and demagogue. Their primary mission will be to rail against the evils of the health insurance reform bill and make loud announcements about how they're going to slay it. There will be lots of talk about making tax cuts for very, very wealthy people permanent in order to help "small business." No matter what happens, they will never let DADT be repealed on their watch. Etc.
But to help further their cause, they'll do what the GOP always does: Sling mud. I don't think even they are dumb enough to try to impeach Obama himself, but there will be lots of talk about it. There will be inquiry upon hearing upon outraged press conference, spiced up with calls for independent counsel investigations. They pull a Breitbart and find some plausibly bad-seeming thing and use it to force some hapless official out of his or her job. Then they'll bay at the moon, lick the blood from their chops, and lope off in search of another victim.