Tim says that the Tate Modern is Ikea, but for art... what can I say..? I like both.
Yes, it is full of tourists, and it hasn't got the atmosphere of a little gallery somewhere in east London, but it is still a beautiful space with interesting exhibitions (most are for free), it has also got one of the best bookshops ever and a great play area for children with a slide which plays music every time you go on it - and how cool is that?
I would recommend going to Waterloo first (which is easy to) and than walk along the river to the Tate.
There is always something on at the Southbank Centre - check their website - for example from 11th February they run a children's festival called IMAGINE - which has got lots of free events and a vintage funfair. There is going be something for all ages, even babies: theatre shows, book readings, art activities, music, dance workshops etc... I did it year ago and it was really good.
Anyway it is a nice walk especially when it's getting dark and you can see London lit-up.
According to Odessa the best thing about Tate Modern is entrance hall because it's huge and perfect for running around especially when you have your friend with you...
I always take her to see some exhibitions and try to explain to her the images with simple words. Of course in the end we are ending up in children area which is small but very nicely designed with above mentioned brilliant musical slide!
I think Tate Modern is great for days out with children.