Well, ultimately, planning the best trip will depend on many things. How long do you want to travel on each method? What do you want to get from it? Etc. All rhetorical questions really for you to think about
For example, you could go from Heathrow on the Piccadilly line up to Kings Cross St Pancras, which will give you a connection with the Thameslink. From KXSTP I'd catch a Thameslink to London Bridge, then a Southeastern service to Woolwich Arsenal. At Woolwich Arsenal, you can catch the DLR back into the city and then from there go to anywhere that a Borismaster runs from. That's just one itinerary you could use off the top of my head, although you may not want to spend so long on the Picc. There are many different combinations, the
TfL London Rail and Tube map is your best friend here. Pretty much all achievable with a Z1-6 Travelcard.
The Borismasters currently run on routes
9,
11,
24,
148 and
390. Some also run on the
38. Maps and routes included to show where you can pick one up.
The last thing I'd add is that each of those you list and intend to visit will be very different depending on what time of the day it is and where you are on them. The Thameslink core will be vastly different in the peak and the DLR will be different the further out of town you are, for example.
If you want anymore specific information about something in particular, just ask away!
