I highly recommend cockatiels as family pets. They so often have sweet, pleasant dispositions. Here's a nice write up on 'tiels as family pets:
Cockatiels: The Perfect Pet? - Food and Home | Wondertime
Not as easy to find but worth looking for would be lineolated parakeets. They are also known for having a sweet disposition when hand-raised, are bigger than a budgie but smaller than a cockatiel and are SO FLIPPIN' CUTE:
Lineolated parakeets
Conures (particularly the pyrrhura family, like green-cheeked, black-capped, maroon-bellied, etc.), lovebirds and parrotlets are also a consideration.