When are you planning to visit? There's an ongoing attempt to get some of the local mustachians to meet up, of we know when you're here we may be able to plan to include you
OK lots of great advice here, as a local here's a few thoughts off the top of my head:
1. No need to pay for a travelcard anymore. With a pay as you go oyster, daily and weekly fare capping nowadays means you will never pay more than you would have done for a travel card - fares add up until you reach the price of the travel card, at which point it stops charging you any more.
2. If you want a west end show, the TKTS booth is the place to go as it is run by the theatres themselves to sell unused tickets. There are other booths that look the same and are generally a rip off. However I think the top trick for tickets is the groundling tickets at Shakespeare's Globe, as mentioned earlier. £5 to see Shakespeare performed as it was inteneded, great atmosphere usually with some degree of audience interaction. I've been several times and always been blown away. Just wear comfy shoes as it is standing, and bring an umbrella just in case!
3. Cheap views of the city - the tate modern has great views over the river to St Paul's from the upper levels and is free, and worth visiting in any case. The Tower Bridge museum is less than £10 and includes a fascinating tour of the workings of the bridge along with a walk across the high level walkways with amazing views - to my mind probably one of the most under-rated attractions in town and worth every penny. Also to the east the emirates skyline has great views and only costs a regular tube fare.
4. Near Tower Bridge is an awesome area; to the north St Katherine's Marina is a beautiful oasis of calm, and has a good food market on Friday lunch times. To the south the cobbled back streets of shad thames feel like something out of a dickens novel, very atmospheric. And the More London scoop is a good place to chill with nice grassy area looking over the bridge and often festivals in the summer.
5. British food is not bad. It also can be very expensive but does not have to be! What is very expensive and bad is food in tourist areas. If you buy food at a restaurant near to any tourist attraction it will almost certainly be bad and overpriced. But maybe a street or two away where the locals eat you can get much better food at a fraction of the price. Here's a good place to get you started:
http://www.timeout.com/london/food-drink/londons-best-cheap-eatsDefinitely worth getting the tripadvisor app, even without mobile data you can download their london guide for offline viewing and it will help you to get good cheap food wherever you are. Another great food market I haven't seen mentioned is Greenwich, they have the best noodles I've ever eaten, spun by hand while you watch.
6. As mentioned don't always trust the tube map. Don't be afraid to take buses, they're not just for juveniles and murderers as I gather they are in much of the US. Because you can see where you go you'll get a much better idea of how the city fits together than you do on the tube - no need for a special tour bus. Also bring comfy shoes and walk a lot, its the best way to experience the city.
7. If you have a smartphone or tablet the citymapper app is free and is an invaluable help for getting around - I use it every day. You can plan a journey when you have wifi at your friends house or at most bars, restaurants, coffee, mcdonalds etc. You can also buy a pay as you go sim to put in your phone - maybe not the most mustacian but could save you a packet. For example you can get a "Three" sim with 30 days unlimited data plus loads of minutes and texts for £15 - if that helps you find a cheaper restaurant or avoid a cab fare because you know where to get a bus instead it could easily pay for itself.