Sprinkles Gelato is horrendous now! Creams on Charminster Road is wayyyy nicer.
Oceanarium on the beachfront, the beach itself (avoid the far Boscombe end and under the pier, junkie needles galore), Bournemouth Pier, loads of soft play places.
Poole- Farmer Palmers in Organford, Poole park, Tower Park (not actually a park but a leisure complex - has a huge soft play place called Lemur Landings too), Upton Country Park (really lovely place), Brownsea Island, Poole Quay, Poole Pottery (not sure if they still do the make-your-own plates etc), museum. Paultons park/peppa pig world is about half an a hour away if you're willing to travel. Moors Valley is about 20 mins away. Avon Heath Country Park is about 15 mins away.
Food (Poole) - loads of food places in Tower Park, Day's (buffet), Corkers on the Quay, Da Vinci's, Rope & Anchor.
Bournemouth - Funki Griller, Starz diner, The Lost Garden, 7Bone, Rancho, Tiien