Manchester is not a beauty spot and does not pretend to be. It was the first industrial city and it still behaves like a city: warehouses, a real centre, universities, and museums that are free and good. Two nights lets you use it properly. One night is a gig and a station.
It is also the rail door to the Peak District and to Liverpool. Sleep here if the north-west is the region and you want streets rather than a village.
See
The Whitworth or the City Art Gallery
Both free. Pick one and stay. The Whitworth has a park. The gallery is in the middle of town.
The Science and Industry Museum
In the old railway warehouses at Castlefield. The right rainy half-day, and honest about how the city was made.
A walk along the canal
Castlefield basins to Deansgate. Short, and more useful than a bus tour.
The Central Library
The reading room is a public room, not a paid set. Go in, even if you only sit for ten minutes.