Hello,
I hope the weather stays nice for you! Weymouth beach is quite long and the water is very shallow for ages making it safe for kids to play in. There are donkey rides and other attractions on the beach. There is a long esplanade which you can stroll along from one end of the beach to the other. Half way along is a crazy golf course and Lodmoor Park where there is stuff for kids to do. There are plenty of shops, some high street stores and lots of souvenir shops etc. There's a cinema and bowling for the kids There are a number of pubs - I would advise walking through the town to the harbour where you will find pubs on the quayside where you can sit in the sun and watch the boats coming in and out. On the other side of the harbour is Brewers Quay with shops and further up is Nothe Fort which has tunnels built under the ground and is related to WW2 (I think). You can get buses from the back of Debenhams which will take you to Portland where there are pirate graves (bit hard to find unless you know google Church Ope), beaches, Portland Bill lighthouse and Portland Castle. You also have a great view over chesil beach. Chesil beach is not suitable for going in the sea with kids as it has a steep drop not very far out and strong currents but is great for fishing. The Cove Inn is on Chesil Beach and does nice seafood.
Nearby attractions (in case it rains) are Monkey World (which is a rescue centre for monkeys and apes), The SeaLife Centre, The Tank Museum, Abbotsbury gardens and Swannery, Dorchester Museum. You could take a day trip to Dorchester town (easily accessible by bus or train). Buses from The Kings Statue can take you to Lyme Regis to hunt for fossils with the kids. You can also get fast ferries to the channel islands from Weymouth. Can't think of anywhere you should especially avoid...have a great time!