(CNN) -

The bill that backs the United States away from its fiscal cliff awaited President Barack Obama's signature Wednesday, but new battles over taxes and spending await Washington in the next few weeks.

Congress averted that self-built precipice late Tuesday when the House voted to stave off widespread tax increases and deep spending cuts by accepting a brokered Senate compromise. It makes permanent the Bush administration's tax cuts for individuals earning less than $400,000 per year and couples earning less than $450,000.

It raises rates on those who make more than that from 35% to 39.6%, bringing back a top tax bracket from the Clinton administration, and will raise roughly $600 billion in new revenues over 10 years, according to various estimates.

  • If you earn $30,000 a year, you will pay $365 in new payroll taxes.
  • If you earn $52,000 a year, you will pay $679 more in payroll taxes.
  • If you earn $84,000 a year, you pay $2 less in income taxes but about $1,1000 more in payroll taxes.
  • If you earn about $250,000 a year, you'll pay about $2,500 more in income taxes and about $3,100 more in payroll taxes.

The bill also extends unemployment insurance and delays for two months the threat of sequestration -- a series of automatic, across-the-board cuts in federal spending.

Economists had predicted the combination of those tax increases and spending cuts could have thrown the U.S. economy back into recession and driven unemployment back into the 9% range.

Meanwhile, a new Congress takes office on Thursday, and lawmakers will soon be confronted by the need to raise the federal debt ceiling and what to do about the still-hanging sequester -- a legacy of the last battle over the debt ceiling, in 2011.

Obama said he would sign the bill into law, but he did not say when. After the vote, he flew to Hawaii to rejoin his wife and daughters on their winter vacation.

"The sum total of all the budget agreements we've reached so far proves that there is a path forward that is possible, if we focus not on our politics but on what's right for the country," Obama told reporters late Tuesday. "And the one thing that I think, hopefully, in the new year, we'll focus on is seeing if we can put a package like this together with a little bit less drama, a little less brinksmanship, not scare the heck out of folks quite as much."

The Bush tax cuts expired at midnight Monday, while sequestration had been scheduled to start when federal offices reopened Wednesday.

Congress planned to send the bill to the White House on Wednesday, a Republican leadership aide told CNN, but there's no urgency on the president's signature in practical terms. It's up to the Obama administration to implement the budget and tax changes, and since the president has said he will sign the measure, the administration can begin planning for the changes immediately.

World markets rose after the late-night vote. U.S. stocks jumped, too, with the Dow Jones Industrial Average rising nearly 2% by mid-afternoon.

Tuesday night's 257-167 vote saw House Speaker John Boehner and about a third of his Republican majority lining up with the Democratic minority against most of the GOP, including Majority Leader Eric Cantor and party whip Kevin McCarthy.

Rep. Nan Hayworth, an outgoing Republican representative from New York, said she was a "reluctant yes."

"This is the best we can do, given the Senate and the White House sentiment at this point in time, and it is at least a partial victory for the American people," Hayworth said. "I'll take that at this point."

The Senate plan was brokered by Vice President Joe Biden and Senate Minority Leader Mitch McConnell, R-Kentucky, and it passed that Democratic-led chamber 89-8. But many House Republicans complained the bill did too little to cut spending while raising taxes for them to support it.

Conservative lobbyist Grover Norquist, whose Americans for Tax Reform pushes candidates to sign a pledge never to raise taxes, said the plan preserves most of the Bush tax cuts and won't violate his group's beliefs.

"The Bush tax cuts lapsed at midnight last night," Norquist tweeted Tuesday. "Every (Republican) voting for Senate bill is cutting taxes and keeping his/her pledge."

But Rep. Jeff Landry, R-Louisiana, told CNN's "Early Start" that Obama convinced Boehner "to undo everything he promised he would do" after the 2010 elections that gave the GOP control of the House.

"They did a debt ceiling deal, gave the president $2.1 trillion," Landry said. "They turned that deal off for two months. That's going to be another fight on top of the sequestration, a debt ceiling fight."

Other Republicans warned that as they did in 2011, they'll be demanding additional cuts before they agree to raise the federal cap on borrowing.

"The president has maxed out his credit card, and he is not going to get an unlimited credit card," Sen. John Barrasso, R-Wyoming, told CNN. "We're going to talk specifically about cuts and specifically focused on tax reform as well as helping to save and strengthen Medicare and Social Security. And that's the next discussion we're going to have in Washington."

The federal government bumped up against its $16.4 trillion debt ceiling on Monday and has about two months before it runs out of ways to shuffle money around to keep Washington within its legal borrowing limit. Obama had sought to resolve the issue as part of the fiscal cliff negotiations, but the issue never made it to a final bill.

Tuesday night, the president warned Congress that he will not tolerate another round of brinksmanship that could have "catastrophic" effects on the global economy.

"While I will negotiate over many things, I will not have another debate with this Congress over whether or not they should pay the bills that they've already racked up through the laws that they've passed," he said.