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The Underground from LHR to London is a good value (L3.50), particularly if you have all zone Visitor Travelcards and begin their use at LHR so this first trip is included. As mentioned above, it will take almost an hour. I don't recommend trying it during peak times with lots of baggage. The Underground is like the New York subway, designed for carrying people with no specific space allocated for baggage. Fine for one or two people each with a single bag, but probably difficult to manage for a family with typical vacation travel bags.

On the other hand, the Heathrow Express is fast and convenient, designed for moving people and their baggage. First class is nice, but not worth the premium over standard class for a 15 minute trip. Adult standard class return is L22, or L20 if purchased on-line. Up to four children (15 years or younger) can travel free with each paid adult. There is also a "Group Weekender" special for four or more adults traveling together, obviously on a weekend (as you plan to). Check out www.heathrowexpress.co.uk. If you're buying BritRail passes for travel outside of London, remember the Heathrow Express is included in the pass.

My recommendation: take the Heathrow Express. Much easier with a family. You'll most likely be tired from the flight. Why start your holiday with a stressful trip into London? Relax on the train; you'll have lots of opportunity to experience the tube later. Sure there will be a cost, but how often does your family travel to London?

Enjoy!