TfL - Transport for London, is likely the best, its official, and (at least when I've used it) accurate. Citymapper and others pull from the TfL feed, I believe. TfL will also pull in bus maps/routes for connections.
Apart from that, an old fashioned paper map. Never worried about dead batteries, loss of wifi or mobile signal. Wont break if dropped, thieves think they are useless, and if something DOES happen to them, they are easily and inexpensively replaced.
If you are looking for something that has almost ridiculously detalied informtation about Tube stations (maps of stations, faciltities offered, station special notices, station history, transport connections, etc) you can try StationMaster, by Geoff Marshall. Mr. Marshall is a two-time World Record holder for the shortest time visiting all 270 Tube stations in a single day, and a bit of a Tube enthusiast. For bus routing and detail, StationMaster pulls its connecntions data from...TfL.
One final suggestion, if I may - please make sure you try London's buses. They are cheaper, and depending on the hour of the day, quicker than the Tube (commute times can make stations and trains absolutely rammed). You also get to see the wonderful city that is London.