Tate a Tate
This was, I suppose, inevitable. To walk from Tate Modern to Tate Britain, though you can take the Artbus.
First the simple way. Along the Thames Path on the South Bank (for as the river turns in an arc, this is the shorter route), along the Bankside, under bridges, past Lambeth Palace and the Museum of Garden History, over Westminster Bridge, along Millbank and there you are.
Now a grander way, but let's do it in the other direction for full effect. From Tate Britain, head up Millbank, lots of traffic but not for long for you soon come to gardens then Parliament buildings and the square. One day all this will be traffic free? Now up Whitehall to Trafalgar Square, also one day to be traffic free, and the National Gallery. Now also the restaurant of the National Portrait Gallery competing with the Tate for rooftop views. Glass of champagne an expensive more than six quid.
Next along Strand to Aldwych, past the remains of the John Adams Adelphi and the Royal Society of Arts. On Aldwych not only the baroque churches but now Somerset House Square a triumph of getting civil servant motor cars out, proving it can be done, the re-opened Terrace, and the Courthauld Gallery.
The from Aldwych along Fleet St. to St Paul's and over the bridge. Rather a lot of traffic on this route but so much to see. Described in lots of walking books.
At almost each stage you can drop down to the river along the Thames Path.