Hi there, I have 3 young children - 8, 6, 5 and am living in St Julians at the moment.
With regard to beaches, for young children I would recommend Armier Bay and Little Armier. The sea is generally calm and is shallow for quite a way out. There are some cafe's very close by and you can get sunbeds and umbrellas to hire. Parking is no problem. If you like to snorkel then Ghar Lapsi is a an excellent place to go, but is quite small with no sand, just rocky ledges, there is also an excellent restaurant at the top by the parking area.
Mellieha Beach and Golden Sands get very crowded so nearby is the beach called Ghajn Tuffieha, which is sandy and less popular because you need to drive to it.
You will need a car to get around as the buses are particularly crowded during the summer.
You might want to check out the possibility of jellyfish in the sea. There have been a lot of problems again this year with jellyfish. My father was stung and several of my children's friends have also been stung. The stings can be treated with vinegar or ice (or urine!) also Waspeze helps (so I've been told).
I hope this helps. If you have any more questions, particularly regarding the children then please let me know!