The Aberdeen-headquartered group, which includes the South Western Railway (SWR) and Great Western Railway franchises, said it will continue to manage its rail operations but warned that it has concerns over "risk and rewards" in the sector.
It added that it is waiting for the outcome of the UK government's review into the rail industry and said "any future commitments to UK rail will need to have an appropriate balance of potential risks and rewards for our shareholders".
The firm - one of the largest operators in the UK with a fifth of the market outside of London - said it believes "now is the right time" to separate out its bus division, which it added has "limited synergies" with its other operations.
It said the formal sale process for Greyhound is now under way in a move to deliver "best value for shareholders".
Unite national officer for passenger transport, Bobby Morton, said: “I want to make it crystal clear that Unite won’t tolerate one single job loss or any attack on our members’ terms and conditions as a result of this sell-off process.
“On one hand, FirstGroup has praised staff for their hard work and commitment and then, in the next breath, it announces it is selling off First Bus because of so-called ‘limited synergies’ between the parent company and the bus operations.
“If any potential buyer for any of First Bus’ operations in the UK thinks that there are rich and short-term easy pickings at our members’ expense, they will need to think again."
Unite has about 20,000 members working for First Bus
The group's break-up plans come as part of a move to focus on its North American businesses - the First Student school bus division and First Transit, which account for nearly two-thirds of annual earnings.
Matthew Gregory, the recently-appointed chief executive of FirstGroup, said: "We see significant potential to generate long-term, sustainable value and growth from the solid platforms these businesses provide in the North American mobility services sector.
"We are intent on executing this strategy at pace, having full regard to the regulatory and stakeholder procedures and approvals that will be required."
The board at FirstGroup has been facing growing pressure from an activist investor calling for changes at the group.
Coast Capital, which holds a 9.8 per cent stake in FirstGroup, had argued that the firm's share price was being held back by less profitable businesses, which was offsetting the stronger First Student unit. It had requested an extraordinary general meeting and called for six of FirstGroup's 11 directors to be replaced.
Details of the overhaul came as FirstGroup posted narrowed pre-tax losses of £97.9 million for the year to 31 March against £326.9m the previous year. It said underlying pre-tax profits rose 13.1 per cent on a constant currency basis to £226.3m.
FirstGroup's results showed the First Student business hiked annual earnings to £173.5m, up from £156.5m. But the Greyhound intercity bus business suffered a plunge in operating profits to £11.4m from £25.5m the previous year as it was hit by competition from low-cost airlines and more Americans using their cars thanks to lower fuel costs.
In the UK, the First Rail division also saw earnings rise - to £72.3m from £57.8m, while its regional bus business posted earnings of £65.8m, up from £50.2m. But the firm booked a £102.1m write-down on its strike-hit SWR business amid "high levels of uncertainty around the franchise".