The group's rail business also won a boost with the award of an extension to a franchise to run trains in south west England and Wales.
In an update on trading in the six months to September, FirstGroup's chief executive Tim O'Toole said he was pleased with the company's performance during a period when it faced "continued economic headwinds" in some markets.
"While it is still early days, we are on track with our plans to return the Group to a position of strength," said Mr O'Toole.
Four months after the Aberdeen-based company completed a £615m rights issue to help cut debts and to maintain its credit rating, FirstGroup highlighted positive developments at the key UK bus business.
The division is expected to report passenger volume growth, of 0.6%, for the first half. FirstGroup said this would be the first volume increase recorded in any six month period since 2008, when the financial crisis triggered a deep global economic slowdown.
"Where changes to our commercial proposition have been in place longest or have changed most significantly, such as in Sheffield, Rotherham, Doncaster and Manchester, we are seeing particularly pleasing increases in commercial passenger volumes," said FirstGroup.
Like-for-like passenger revenue in the division is expected to increase by 1.6%.
In the rail division, like-for-like passenger revenue is expected to increase by 5.7%. The division operates four UK franchises including ScotRail.
FirstGroup suffered a costly reverse last year when it was stripped of the West Coast Main Line franchise it was awarded in the summer when failings were discovered in the bidding process.
Mr O'Toole said the agreement to extend the First Great Western franchise for a further 23 months to 20 September 2015 was good news for passengers, taxpayers and the group's shareholders as it provides continuity and consistency, building on the improvements it has already delivered.
He added: "We remain committed to maintaining a leading position in the market, and look forward to the rail re-franchising programme gathering pace in the coming months."
FirstGroup intends to bid for the Essex Thameside and the Thameslink, Southern and Great Northern franchises, which have been put out to tender.
The company said it "continue to work through" its recovery plan for the student transit division, which operates 50,000 yellow school buses in America. The company continues to focus on winning or retaining only those contracts that meet its returns criteria.
FirstGroup said the core maket for the Greyhound inter-city bus operation "continued to feel the effects of the prolonged US economic downturn, with like-for-like revenue expected to be down 2.4% for the six month period".
It welcomed tentative signs of improvement over the summer.
Shares in FirsGroup closed down 3p at 123.7p.