Lots of good advice in the posts above. The family and I spent a week in London this past summer (me + DW + 5 kids (tweens thru teenagers). Some thoughts from me (apologies if it repeats info from some of the above posts).
I did a bit a research on transit cards and for us, the 7-day London TravelCard worked out best. Get a 7 day pass from a National Rail Station, and you get 2 for 1 discounts on a number of sites (You have to purchase from a National Rail Site - getting from a Tube station will NOT work to get the 2 for 1 deals). We felt it was better than the Oyster card, and we didnt want to get locked in by purchasing one of the all access passes. We definitely got our monies worth on the transit part - took buses/tube everywhere. Used the 2 for 1 a couple of times which helped. Another note that if you do purchase this type of TravelCard, you will need a passport sized photo.
As someone mentioned, if you have a smart phone, there is a great London transit app. We stayed by King Cross station and could put in our location and destination and it would let us know which bus, tube, etc. to take and the location of the bus stop. Could watch on the map in real time where you were going.
We thought about going to Stonehenge, but after some back and forth, decided that we would make a day trip to Bath. Not quite as easy to get to Stonehenge and my understanding is that you cannot go directly up to the monument. You are kept some distance away. Bath is highly, highly recommended. All the kids loved it - great scenery, great history, etc. Definitely one of the highlights of the trip. We took the train there. I would have to double check but I believe it was around $35/round trip or so. ~ 1.5 hours one way.
Warner Bros. studio was another day trip. If there are Harry Potter fans, again, recommended. We did it ourselves - train to bus and was quite easy. There are all inclusive tours offered (transport + entrance to the studio), but with a bit of research, you save quite a bit (on the transport) by doing it yourself. We spent the whole day there (though it does take about 1.5 hours to get there).
Lots of free museums. We did do the Jack the Ripper walk. Guide was great, but the city has changed so much since that time that you really have to use your imagination to see how things were. Relatively inexpensive things we did and enjoyed: Tower Bridge, The Monument (which has a vertigo inducing circular staircase to the top!), Shakespeare Globe theater (not so cheap but very interesting), the Clink Museum
We did not rent a car, and did not have any need for one. The travel pass and taking the train for the day trips worked out just fine.
We rented an apartment via Homestay. 7 people and worked out just fine, but we live together all the time, so are used to each others company. LOL
All in all, a great week. Kids are already asking when we can go back. Any questions, let me know.