Either get an Oyster card for each person in your party, or if you have a 'contactless' bank card you can use that instead. That will get you the use of all the public transport - buses, tubes, DLR, overland, etc.
Trip 1 - river bus from Embankment down to Greenwich. In Greenwich you can see the Observatory, Naval College or Cutty Sark and there's a street market, and then either go back the same way, or take the cable car across the river and catch the DLR back to central London.
Trip 2 - Portabello Road Market - Friday and Saturday are the busiest days.
Trip 3 - British Museum and Covent Garden. the Soane Museum is also close.
Trip 4 - Natural History Museum / V&A / Science Museum - pick one as that will be enough for a day, then head up to Hyde Park / Kensington Park.
Trip 5 - Globe Theatre, Tate Modern, St Paul's Cathedral - all are within easy walking distance of each other. If you visit St Paul's try to go up the dome.
I would avoid the big tourist attractions like Madame Tussauds, London Eye, etc, unless you absolutely want to, as they are very expensive and have massive queues.
Unfortunately, as with all big cities you will need to watch out for pickpockets, particularly in more touristy places like Portabello Road or at the busier tube stations or bus stops. Make sure you keep your bags to hand in restaurants as well. London is not a den of thieves but there are always opportunists who will prey on the most naive. Travel light in terms of camera kit as the above will still entail a fair bit of walking, and you'll be carrying it with you. That's all I can think of for now.
