Trentbarton says the rises will take effect from Sunday, April 10, affecting the Rainbow One service which runs through both towns.
The company says many fares are being frozen and some groups of customers will see no rises at all.
But faced with high inflation and increasing costs of fuel, parts and wages, fares across several routes will be increasing.
Company bosses added, though, that travelling by bus remains the green option to cut carbon and air pollution, and will still be cheaper than the equivalent solo journey in a car.
Trentbarton’s Mango app is increasingly the cheapest way for bus users to travel, with discounted singles, new zonal caps and rolling caps.
Customer feedback has called for a simpler and clearer fare structure and Trentbarton said is responding by making Mango a first choice for both frequent and irregular users.
The Mango off-peak day cap will be £6.30, while the Mango peak seven-day cap will be £10.10, and the introduction of zone seven-day and 28-day caps to Mango enables customers to access cheaper caps without having to do anything differently.
Single tickets – including those paid for using contactless bank cards – remain an important way to get people travelling by bus and any increases to singles are being minimised.
To encourage couples, families and friends to travel together, the duo and group Zigzag prices are remaining frozen, as is the £3.50 Zigzag-plus add-on which includes morning peak travel.
Trentbarton says it will use the increased revenue from the fare increase to not only pay for fuel and and staff wages, but to continue to invest in its services and fleet.
Tom Morgan, group commercial director at Trentbarton, said: “We only increase fares when we have to, and the current inflationary pressures on our costs have to be covered for our services to remain viable.
“We are also simplifying our fare structure after listening to customer feedback and accelerating improvements which make our most popular ways to pay the best value for money.
“Keeping our single fares attractive means we are the great value and green choice for occasional travellers keen to get on board with the best way to travel for people and their environment.”
A full guide to the fare changes is here.