It is impossible to book Manchester to Bournemouth flights because Bournemouth Airport offers no domestic services, and only operates flights to and from continental European destinations.
The nearest airport to Bournemouth which does offer domestic flights is Southampton however and it is very easy to travel between Southampton Airport and Bournemouth. This part of the journey is discussed later.
Flybe (http://www.flybe.com) offers multiple cheap flights every day throughout the year from Manchester to Southampton, with relatively consistent costs. The flights take approximately 1 hour in the air.
Assuming you book more than a few weeks in advance, the vast majority of the one-way flights will cost £40, and return flights £80. Those flights with a higher demand push the one-way ticket prices up to anywhere between £50 and £70 and the return tickets up to between £100 and £140.
If you need to travel with shorter notice, it is often still possible to find £40 one-way tickets and £80 returns if you have a little bit of flexibility regarding the exact date, but it might be necessary to fork out between £60-£110 each way if your requirements are more rigid.
Upon arrival in Southampton, a regular train service from Southampton Airport Parkway to Bournemouth can cost as little as £2 if you buy online from Cross Country via http://www.nationalrail.co.uk. The trip takes around 40 minutes, and there are trains throughout the day close to the arrival times of all the flights from Manchester.
Using this journey plan, it is possible to travel from Manchester to Bournemouth via flights and a simple rail trip with a total journey time of less than 2 hours and 30 minutes.