The laws on public photography are clear. If you take many photographs it's even worth carrying a copy given that people are so paranoid these days and the police aren't always clued up.
In essence, in a public place you can photograph anyone, including children, and use those images both privately and commercially. The only provisos are that you don't harass the person(s) you're photographing - for example jumping out of bushes and following them around (especially if it happens on more than one occasion) - and you're not doing something that's clearly "weird", like spending the entire day photographing the childrens' play area.
There is a grey area in terms of privacy, but that doesn't apply in a public place, only to locations where you wouldn't expect someone to be taking photographs (in the doctor's waiting room, the tanning salon etc).
The Home Office minister said: "There is no legal restriction on photography in public places, and there is no presumption of privacy for individuals in a public place."
Bear in mind however, that in places like zoos, you have to observe their own terms and conditions which are often placed on society members and as part of the contract when buying a ticket. For example as this is a Chester thread, anyone can take photographs at Chester Zoo, but they cannot be sold or used commercially. If you wish to make money from your photos of Chester (and other zoos), then you need to pay the zoo for a license, and then they might want to vet what you shoot. You can't print your photos on T-shirts and mouse-mats, unless you plan to give them away.
As Google will tell you, property owners can't stop you taking photographs of their home or business provided you're standing in a public place. They most certainly can't ask you to delete your images, and the police can't either.