But staying for 3 weeks in August, before booking airfare you need to find somewhere to stay in the Lake District for 3 weeks- given that it is the last day of June now, that could be a challenge, as August is peak season in the Lake District.
Are you thinking of a B and B or a self catering place?
I'm from the Lake District so am aware of how busy it gets.
You may need to get a little bit creative and stay in Kendal (at the south end of the Lakes but only half an hour from Windermere by very frequent bus) or Cockermouth on the west side of the Lake District- but only half an hour from Keswick on a half hourly bus.
Before you book somewhere check it is on a bus route!
This is the Lake District summer timetable- https://tiscon-maps-stagecoachbus.s3.amazonaws.com/Timetables/Cumbria/Lakes%20Connection/Summer%202023/CNL%20Summer%2023%20Lakes%20Guide%20WEB%20V2.pdf
You may possibly have to relocate each week.
While bus fares are on a £2 cap at the moment the best value bus ticket for each of the 3 weeks is a North West Gold 7 Day Megarider for £34 each- which works out at £5/day- so you only have to do 3 bus trips each day to make it pay. Each time you change a bus is one £2 fare.
All buses in the area that you will use are run by Stagecoach.
As there are only two of you that is the cheapest option- if there were three or more you would want a Group Megarider, like I suggested to a family of 4 earlier today.
To forestall someone trying to argue what area that ticket covers it is the entire Lake District, and to Newcastle, Dumfries, Liverpool, Chester, Skipton and Bolton (near Manchester) although some of those are very long day journeys which a tourist would not do.
For train times from Manchester Airport to Windermere and the Cumbrian Coast you want timetable 6 on this web page- https://www.northernrailway.co.uk/travel/timetables
For Manchester Airport to Penrith direct see here- https://timetables.tpexpress.co.uk/#/timetables/2304/Anglo-Scottish
Depending on where you end up staying decides which rail head you use from Manchester.