All I can say is that I really enjoyed a weekend in Liverpool some time ago. The links above all offer really interesting things to visit if you like that sort of thing.
And for shopping the central area Liverpool One has endless streets of shops
Unless you particularly want a big city (Manchester or Liverpool) why not consider Lancaster? Much nearer the Lakes - within an hour's drive so you could easily have a full day there. There's the Castle (former prison & court), the Ashton Memorial in Williamson Park (the memorial is one a high point, visible for miles around and on a clear day offers great views), a couple of museums, the canal & river for walks, etc.
If you want a traditional seaside resort, Morecambe is only a short hop from Lancaster so you could visit there as well.
I noticed in another post that you will have a hire car. Of the cities you mention I would add another vote for Liverpool. It will probably take nearly 2 hours to drive there though, and once on the M6 motorway it will not be a very nice drive. Another big city you could consider is Newcastle, the drive there is more scenic, especially if you take the route along Hadrians wall, although the drive will be even longer. I would also avoid Blackpool.
Another vote for Liverpool. Fabulous, lively city, beautifully regenerated with plenty to see. A bit of a drive from the Lakes but a good rainy day alternative.
On a far smaller scale but another pleasant excursion is Carlisle.
Liverpool ! I lived there and worked in the lakes you can do the journey in a little over an hour if you avoid rush hour, come off the M6 onto the M58 until the end of the motorway at Aintree, park at Aintree station (free) and get the train into town it takes 13 mins and costs a little over £3 return, Have a great day
Great idea to park at Aintree, but it will take 30 minutes to just get onto the M6 from Ambleside, and another 1 hour down to Aintree.
What if I choose to take a train? Is it a better choice?
The nearest train station to Ambleside is at Windermere. It is easily reached, just next to the main A591, a 10 minute car journey away. There is also a regular bus service. There is a car park at the station where you can park all day, but it is not very big and will probably fill up early in summer.
The train journey to Liverpool will take slightly longer than driving, over 2 hours, as you will have to change trains twice. Firstly at Oxenholme (near Kendal), then Wigan. At the moment there is one train late morning which goes direct to Wigan so you would only have to change once, but it would not leave you much time to explore.
Manchester is more easily reached by train, it is not quite as far and you only have to change trains once at Oxenholme. The same late morning TransPennine train goes direct to Manchester Piccadilly. There is a very good Metrolink tram system in Manchester which makes it easy to get about.
It is possible the train timetables will be different in summer, so you would have to check nearer the time.