Just get up, grab a picnic again and just walk! leave the kids to direct you and you could honestly end up anywhere.
Buy a Bus Ticket and just get on any bus, In East Kent you buy an explorer ticket and the children travel for 10p each, this makes for a really cheap day out and you can stretch the day from morning to night.
Take a bus to the nearest bus station then let the children choose which bus they want to go on next, Then again when they want to change bus, just get off and hop onto another bus. Think of the places you could end up and the different things you could see.
Better still if you do it whilst walking you can get loads of exercise done and it will be so good for you all :)