or bus number 243 from back of station to Old St Station takes about 25 mins - easy if you have a buggy, unless it's packed you can wheel it straight on. If you have older kids, tube as above is easy. The website TFL is great for getting around London.
More direct would be to take the Waterloo and City Line from Waterloo to Bank (that's the only place it goes - it's a shuttle between the two stations). Then it's just two stops on the Northern Line to Old Street Station. Allow 20 mins. W&C Line doesn't run late evenings during the week, after 6pm or so on Saturdays, and not at all on Sundays though.
If you aren't a tube fan the 243 bus goes all the way there and goes through such interesting parts of London. The journey will be about 25 mins, and they're pretty frequent, but I'd allow 35 mins from your arrival at Waterloo all the same.