I'd definitely take the shinkansen from Tokyo Station for two reasons: The first reason is that not all shinkansen stop at Shinagawa. If you're using the JR Pass, you might have to wait up to an hour for the next available Hikari at Shinagawa Station.
The second reason is that you'll have more time to aboard the train at Tokyo Station since it's the starting station. At Shinagawa, trains stop only for two minutes and you'll have to rush to get kids and luggage onto the train.