Grab a Tube map, carry it around and study it. The maps are small and are normally offered for free at most Tube stations around London. Some people even collect them, as they tend to feature interesting cover art.
Plot your itinerary in advance and try to take as fewer subway lines as possible. The London Tube system is made up of various subway lines, all named and color-coded. The less subway line changes, the faster you'll get to your destination, because the station where you would be changing to another line may be enormous. Some are several stories high and contain mini-malls!
Stay to your left -- when standing on line, when navigating the hallways, when getting on a train and when standing on an escalator! This is England -- people drive on the left and they walk on the left. Don't stop pedestrian traffic underground or clog up the escalators/stairs, as some people are in a hurry and wish to pass you. This way, you and everyone else will get to where you need to go faster.
Remember the direction you're heading! Getting to the right platform -- northbound vs. southbound trains and eastbound vs. southbound trains -- can be confusing. Look at your Tube map, or a Tube map on the wall, and take note of the last stations on the line. The train you're meant to get on with have the ending station that's in the direction you're heading projected on a screen on the front, and it may also be announced on a scrolling board on the platform. Also, in many cases, the conductor will announce something like "this is a southbound train, heading toward xx." Pay attention.
Mind the gap! An automated voice will repeat this on the train after it arrives at a station. There is a reason for this -- there will often be a very large gap between the train and the platform. So mind the gap and ensure your children, your pets or your luggage don't fall through either!
