By Steve Holland

CHICAGO (Reuters) - President Barack Obama urged Democrats on Thursday to help him "finish the job" at the first events of his 2012 re-election bid, appealing for higher taxes on the wealthy and a rejection of Republican budget policies.

Obama, seeking to reignite the energy of supporters that propelled his candidacy in 2008, said "extraordinary progress" has been made during his two years in the White House but "we've still got work to do."

"If you're just as fired up now despite the fact that your candidate is a little older and a lot grayer, then I have every confidence that we're going to be able to finish the job," he said.

The president used Washington budget battles as a backdrop at the first of three fund-raising events that were estimated to raise $2 million for his campaign from a hometown crowd that included many key figures in Obama's rise to power.

They were the first fund-raisers in what will be a long slog toward a goal of perhaps a record war chest of $1 billion for 2012.

Obama leads a host of potential Republican challengers in public opinion polls, but most surveys give him less than 50 percent support in a sign that his re-election could be difficult.

Obama begins his re-election bid just having avoided a government shutdown in a last-minute budget deal with Republicans, and it was clear from his remarks that there are still strains from that battle.

The Republican approach, he said, is that "we can't afford to do big things anymore" and says to the underprivileged, "tough luck, they're on their own."

Obama, who reluctantly agreed to extend Bush-era tax cuts late last year even for wealthier Americans, said if the wealthy were to "pay a little more in taxes," it would help solve America's fiscal challenge without forsaking its responsibility to its people.

"If we apply some practical common sense to this, we can solve our fiscal challenges and still have the America that we believe in. That's what this budget debate is about and that's what the presidential campaign is going to be about," he said.

Obama has attempted to straddle a middle ground and sought compromise with his political adversaries since Republicans took command of the House of Representatives and picked up strength in the Senate at the November elections.

This has led to some strains with his liberal allies, most significantly when he reached a compromise in December with Republicans that led to a two-year extension of Bush-era tax cuts for wealthier Americans.

Perhaps Obama's most serious rival at the presidential election, former Massachusetts governor Mitt Romney, threw his hat into the ring this week by announcing he was looking into running for the Republican nomination.

(Editing by Eric Walsh and Mohammad Zargham)