Although the tube is extremely busy in the morning in rush hour (about 7am until 9.30am), and quite busy in the evenings (about 5pm-7pm), when services are running normally things get a lot less busy outside that window, and some seats will usually available (although not necessarily three together). Unfortunately most lines have cars which don't have passenger accessible connections, which means that if the carriage you're waiting at happens to be full, you can't walk down the train to find an empty one (the new District and Circle line trains do allow this, as well as having working cooling systems, on the other hand it's not the more reliable service in the world). With enough time, it's possible to learn which bits of a train tend to be busy at a particular station, but that's probably not useful for you.
In general, things are a bit easier if you're heading out of central London in the mornings, and into town in the evenings, only crossing the centre near the middle of the day, but that would limit you to West London to a certain extent. If you have a lot of free time (i.e. over an hour) then the London Overground forms an "outer loop" which would usually be significantly less crowded than a trip through the centre, but obviously slower.
Finally, the Tube map has won awards for its clear design, but it lies when it comes to how close different stations are to each other. If you're making a short trip in the centre, check a street map, or else you can end up walking further to reach the platforms than you would to get there on foot above ground.
In terms of not annoying the locals:
- Stand on the right hand side of escalators, and watch out for people walking up on your left hand side
- If there's a queue at the ticket barriers, get your tickets out before you get to the front (and take your child to the wide manned barrier if there is one, since they'll be travelling free).
- Some buses (the ones with three sets of doors, including a single door at the very back) have oyster card readers on each door, so you can get on these ones anywhere. If the bus has only has two sets of doors, the card reader is with the driver at the front and you'll need to get on there.