For eating/activities with kids: Food Hub at Broomfield Hall, Bookcafe, Little Eaton Garden Centre (play area, pet section), Croots (outdoor play area), The Kitchen Sadlergate, The Apple Tree, Bluebells (Ice Cream parlour, farm and play area), Sticky Fingers (pot painting cafe).
Good selection of activities advertised in Lots for Tots free mag.
With/without kids - lovely one-off cafes and handmade stuff advertised on Fabulous Places website. Bottle Kiln, Fresh Basil etc
Outside space: As other posters, plus Attenborough Nature reserve and cafe and lots of English Heritage places nearby to get value from membership - Bolsover, Hardwick, Asbby etc.
Free and cheap stuff: my kids don't tire of Derby museum and Pickfords House. Kids events at Calke have been ok.
Culture with/without kids: Derby Quad, Buxton, Derby Playhouse improving
What I like: proximity to regenerating city (Derby) and lots of nice little towns (Ashbourne, Belper, Castle Donnington, Duffield etc). Plus nearby bigger places - c'mon, John Lewis at Nottingham is fab and close enough! Westfield has its merits for convenience. Great countryside on doorstep too.
I came back to Derby via Yorkshire. DH is northerner by birth, traveller by history and midlander by preference - I quite like that Derby has a bit of everything. Not too north, south, east or west. Posh bits and down at heel in some places but overall on the up I'd say.
What I don't like: lovely lovely people but the place can be insular with so many people working for the bigger employers.