Alnwick Gardens is another favorite of ours. If you plan to visit more than once, make sure you get annual pass (by becoming a 'Friend') which makes it a fair bit cheaper and you get free parking! The little ones love ambling through the gardens and getting lost in the bamboo maze.
Be sure to visit the 'Poison Garden' and find out about the dangers lurking in your own back garden!
You might also want to take a towel and some spare clothes for any children who wish to venture into the 'water garden'. A popular option for the young (and some old alike) is dicing with the unpredictable fountain at the far end which will, without any warning, suddenly drench those within reach!