WASHINGTON — After taking a largely symbolic stand on Wednesday, Republicans on Thursday will begin a new phase of their effort to overturn the sweeping 2010 health care law, pursuing a variety of strategies: court tests, funding cutoffs and piecemeal changes.

The GOP-led House of Representatives voted Wednesday 245-189 to repeal the law, but that effort is likely to go nowhere in a Senate still ruled by Democrats, and even if it passed there, repeal wouldn’t survive a certain presidential veto.

That’s one reason why on Thursday the House plans another vote directing its committees to look for specific changes they can make to the health care law.

Changes over the next two years could involve reducing paperwork burdens on businesses, permitting the sale of coverage across state lines, denying the government funds to implement the law, and denying funds for a series of grants and other health-related programs.

Still, many political hurdles stand between House Republicans and success in those endeavors, as one house of Congress generally needs to reach compromise with the other house — and the president — to achieve anything.

Senate Republican Leader Mitch McConnell of Kentucky said the GOP will push for repeal despite the seemingly long odds, since Democrats control 53 of the 100 seats.

“”The Democratic leadership in the Senate doesn’t want to vote on this bill,”” McConnell said after the House vote. “”But I assure you, we will. We should repeal this law and focus on common-sense steps that actually lower costs and encourage private sector job creation.””

Republicans may have a better chance of success in court.

Some 25 other states have joined Florida’s lawsuit in federal court challenging the health care law — six signed on Tuesday — and Virginia is pressing a separate case. All those states echo a key Republican argument in contending that the law’s requirement that nearly everyone buy insurance by 2014 or face penalties is unconstitutional.

In December, federal District Court Judge Henry Hudson ruled that a person can’t be forced to buy coverage; the Obama administration is appealing. But Virginia, which filed the suit, also is appealing, saying that Hudson should have overturned the entire law. The case is expected to wind up before the U.S. Supreme Court.

First, though, some Democrats in Congress say they’re open to some change: “”We will certainly look at any good ideas that come down,”” said House Minority Whip Steny Hoyer, D-Md.

But they also warn that the law is a carefully crafted balancing act, and removing one piece could jeopardize the success of another.

For instance, Republicans are eager to overturn the individual mandate, but would keep the terms barring insurers from rejecting people with pre-existing conditions.

Doing that, though, risks sending premiums higher, since in theory healthy people would be less inclined to buy coverage while the number of people needing coverage would increase.

“”The argument for a mandate is that if you’re going to lower costs, improve access and improve the quality of care, you have to increase the risk pool,”” said House Minority Leader Nancy Pelosi, D-Calif.

Several bipartisan discussions are under way on Capitol Hill about possible changes in the law, but without strong support from the administration they are expected to languish.