We have something similar to this that my mum bought for my kids.
[
www.scandiborn.co.uk]
It's good because we encourage them to play with the lego on the play mat then you simply use the drawstring sides to tidy up and slide into a sturdy sided toy box. We also use low flat plastic boxes to pack away under beds and sofas.
I wish I was more organised and kept the mechanical pieces separate but all we've managed is to put the lego people into a separate bag.