I'd stay in the city centre. 12yo will love kick air (trampoline and football/sports/adult softplay) type place at the back of Victoria Station opposite Manchester Arena. There is also a VR arcade off market Street near boots but down side street for entrance. Go to China Town for lunch Red Chilli is good. Head to northern Quarter and go to the manga and comic book stores (travelling man?) On Oldham Street near Afflecks palace.
For culture there is Manchester art gallery, ryalnds library, MOSI, the Whitworth just a short bus ride away is a fab contemporary gallery and you can eat at Cro or just the Whitworth cafe and there's a nice park next to it. The Manchester museum is cool. I love it. Its a well laid out natural history museum. My kids now prefer paid for delights rather than free museums.
Or do a trip to the trafford centre by tram. There is a Dino golf, ski slopes, laser quest, "paradise golf", namco station bowling and arcade, ifly, cinema, yo sushi or massive food court and tonnes of restaurants. There's also a massive primark.
Imperial War museum north has just been recommended by my 13yo who went 2 years ago. It's at the lowry which cam also be reached by tram and is a great spot to visit to walk around the quays, shop in the outlet mall (I don't rate it), and get food at the new swanky food court or stick to wagamamas, nandos, cafe rouge or alchemist. There's also coronation Street tour around there!
Don't forget the city or United football stadium tours (not for me but we are famous for football.
Liverpool is lovely but you're coming to Manchester so just do that!
Cheshire oaks is rubbish.