Whatever one feels about Manchester as a place to live or even visit, it's undoubtedly a great base for visiting quite a few other attractive - or at least interesting - destinations in the region. Road links are plentiful and public transport - including rail - quite surprisingly good in this populated and industrialised region of England.
Among the most obvious destinations for day trips from Manchester are:
-- Liverpool. This is the other great city of the North West and although it has as much of a dark side as Manchester does, it has, perhaps, more redeeming features of which the greatest is it's location nearer to the sea. Even if you don't want to partake in the Beatles heritage industry, Liverpool's maritime heritage and handsomely regenerated areas are worth a look.
-- Chester. An ancient town (technically a city, I think, but not a big one) on the Welsh border going back to Roman times, it has a charming if twee centre full of half-timbered buildings, a fantastic circle of medieval walls surrounding the city centre - the best walk in town, Roman remains including a hypocaust (central heating system), one of those brilliantly impressive English cathedrals, a hymn soaring into the sky and an ornate, ugly and very popular Victorian Jubilee clock to take a picture of yourself under.
-- The Peak District. Astonishingly close to the sprawl of Manchester, the expanses of the Peaks (there are two, the Dark one and the White one, and quite different they are too) give a fantastic opportunity for some good hill walking, taking in the landscapes of ancient hills, moody moors and weathered rock formations. The popular long-distance path of the Pennine Way starts in the small hamlet of Edale (which can be reached from Manchester by train) but there are plenty other attractive walks in the area.
Further out, but doable if you have a car at your disposal, are the Welsh mountains of Snowdonia (we did Snowdon from Manchester as a day trip), the Forest of Bowland and even the Lake District and the Yorkshire Dales.