Dylans is also well posh.
There's good fish and chips, pancakes etc in Beaumaris, along with a reasonable playground to let them loose. On the mainland, plenty of eateries around Caernarfon and Llanberis. Dinorwic lodge on the way to a slate quarries walk perhaps.
There's a cafe in the art gallery in Llangefni, although not been there yet, but the gallery is free, and the dingle walk nearby is pushchair friendly (Google dingle Llangefni)