Lots of good advice here. Some points:
In some states (specifically New York) it is actually illegal to transport a child less than one year old by bicycle. One year seems to be a sensible cutoff.
I never felt the need for a kickstand with a trailer, the hitch pivots and I just flop the bike on the ground. However, with a trail-a-bike I'm always looking for something to lean against while we load; if a kickstand could hold the whole rig it would be very handy.
Kids love bikes, just don't overdo it, keep it fun and build up slowly. A couple of weeks ago I did a 20-mile ride on the tandem trail-a-bike with my five-year-old twins, and when we stopped for a water break one said: "Why are we stopping so much? After this, no more stopping for the whole trip!"