The big one. This is where most people will come? Not only for the Tate but for London Eye. It is mis-represented on all the transport diagrams for Waterloo and Waterloo East should be regarded as a single interchange then you have a line running from Dover to Bournemouth with one interchange and symmetry around London Bridge and Clapham Junction, Surbiton and Orpington, Woking (or Guildford) and Sevenoaks, and so on with lots of other interconnections dropping off, such as Cannon St and Charing X.
The complete Waterloo is dealt with and described elsewhere. All we are concerned with here is the Tate connection for which the simple route is through the underground shambles to the river and along the Thames Path. There are none of the little maps here which bless Blackfriars. But there are large posters showing the Bankside promotion which give you a good idea of what is where and you should be able to pick up the Bankside Millenium leaflet somewhere.
Not practicable, I'm afraid, at this stage. Though LT is trying to promote them, they are a joke. They have produced a diagram on which they worked five months, they say, and of which they are very proud.